The linear nature of pseudowords
Jorge Almeida Alfredo Costa José Carlos Costa Marc Zeitoun





  Given a pseudoword over suitable pseudovarieties, we associate to it
  a labeled linear order determined by the factorizations of the
  pseudoword. We show that, in the case of the pseudovariety of
  aperiodic finite semigroups, the pseudoword can be recovered from
  the labeled linear order.


3D Scanning System for Automatic High-Resolution Plant Phenotyping
    
Chuong V. Nguyen
ARC Centre of Excellence for Robotic Vision
 Research School of Engineering, Australian National University

Canberra ACT 2601, Australia

Email: Chuong.Nguyen@anu.edu.au
Jurgen Fripp
CSIRO Health and Biosecurity

Australian eHealth Research Centre

Herston QLD 4029, Australia

Email: Jurgen.Fripp@csiro.au
David R. Lovell
Electrical Eng. & Computer Science

Queensland University of Technology

Brisbane QLD 4001, Australia

Email: David.Lovell@qut.edu.au
Robert Furbank
CoE for Translational Photosynthesis
 Australian National University 
 
Canberra ACT 2601, Australia

Email: Robert.Furbank@anu.edu.au
Peter Kuffner, Helen Daily, Xavier Sirault
CSIRO Agriculture and Food

High Resolution Plant Phenomics Centre

Canberra ACT 2601, Australia

Email: Xavier.Sirault@csiro.au

    December 30, 2023
==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



 




§ INTRODUCTION



Since the publication of Eilenberg's textbook <cit.>,
a large body of finite semigroup theory is in fact the theory of
pseudovarieties of semigroups. Besides its own mathematical interest,
it draws motivation from the connections with computer science through
Eilenberg's correspondence between pseudovarieties of semigroups and
varieties of regular languages. As pseudovarieties are classes of
finite semigroups, only in very special cases do they contain most
general members on a given finite set of generators, that is relatively free
semigroups, namely semigroups on n generators in the pseudovariety
such that every other member of the pseudovariety on n generators
is their homomorphic image. To obtain relatively free structures,
one needs to step away from finiteness into the more general framework
of profinite semigroups, and indeed such a tool has been shown to lead
to useful insights and has found many applications
<cit.>.

As topological semigroups, relatively free profinite semigroups S over
a finite alphabet A are generated by A, which means that elements of S are
arbitrarily well approximated by words in the letters of A.
Thus, the elements of S may be
considered a sort of generalization of words on the alphabet A, which
are sometimes called pseudowords.  Of course, S may satisfy nontrivial
identities, which means that different words may represent the same
element of S, although in the most interesting examples of
pseudovarieties, this is not the case.  Now, words on the alphabet A may
be naturally viewed as A-labeled finite linear orders, a perspective
that underlies many fruitful connections with finite model theory
<cit.>.
For some pseudovarieties, such as
R, of all finite R-trivial semigroups, and DA, of all finite
semigroups in which the idempotents are the only regular elements,
representations of the corresponding finitely generated
relatively free profinite semigroups
by labeled linear orders have been obtained and significantly applied
<cit.>. The purpose of this paper is to
investigate such a linear nature of pseudowords for pseudovarieties with
suitable properties. Our main motivation is to understand pseudowords over
the pseudovariety A, of all finite aperiodic semigroups.

The key properties of the pseudovariety A that play a role in this
paper are of a combinatorial nature: the corresponding variety of
languages is closed under concatenation and the cancelability of first
and last letters. The first of these properties entails a very useful
feature of the corresponding finitely generated relatively free profinite semigroups,
namely equidivisibility, which means that different factorizations of
the same pseudoword have a common refinement. This condition already
forces a linear quasi-order on the factorizations of a given
pseudoword, and this is the starting point for the whole paper. The
cancelability condition leads to special types of factorizations,
which we call step points, to which a letter is naturally associated.
The corresponding linear order has interesting order and topological
properties, such as being compact for the interval topology. The step
points are the isolated points and there are only countably many of
them. All other points are called stationary and, in contrast, there
may be uncountably many of them. Perhaps somewhat surprisingly, there
is no correlation between the number of stationary points and how low
pseudowords fall in the J-order.

Our main result is that the linear order of factorizations with
alphabet-labeled step points provides a faithful representation of
pseudowords over A. We also obtain a characterization of the
partially labeled linear orders that appear in this way, albeit in
terms of properties involving finite aperiodic semigroups. A natural
goal for future work consists in looking for a characterization of the
image of the representation which is independent of such semigroups,
as has been done in the case of the pseudovarieties R and
DA <cit.>.

While this paper was being written, Gool and Steinberg developed a
different approach on the pseudowords over A, applying Stone
duality and model theory to view them as elementary equivalence
classes of labeled linear orders <cit.>. They
worked specially with saturated models. In our paper, the models that
appear in the image of the representation are not saturated in
general.

We also mention the articles <cit.>
and <cit.>, where labeled linear orders were
assigned only to a special class of pseudowords, the ω-terms,
and were used to solve the word problem for ω-terms in several
pseudovarieties, either for the first time, or with new proofs, as in
the case of A, treated in <cit.>.

The paper is organized as follows. After a section of preliminaries,
Section <ref> introduces the key notion of
equidivisible semigroup in the context of relatively free profinite
semigroups, with an emphasis on pseudovarieties closed under
concatenation. Several results of the paper apply to all such
pseudovarieties, but at a certain point our hypothesis restricts
to A. In the next four sections, we develop more on the tools and
the language necessary for the main results. In
Section <ref>, we give our faithful representation
of pseudowords over A as labeled linear orders. The following
three sections relate to the proof of this representation (the first
two of them having independent interest). This is followed by a study
of the effect of the multiplication in the image of the
representation, and by a characterization of the image. The paper
closes with Section <ref> where, among other
things, it is shown that the ordered set of the real numbers can be
embedded in the ordered set of the stationary points of a pseudoword
over a finitely cancelable pseudovariety containing LSl. This
is done via a connection with symbolic dynamics.



§ PRELIMINARIES



We assume some familiarity with pseudovarieties of semigroups and
relatively free profinite
semigroups <cit.>.
For the reader's convenience, some notation and terminology is
presented here. The following is a list of some of the pseudovarieties
we will be working with:


  * I: all trivial semigroups;

  * S: all finite semigroups;

  * A: all finite aperiodic semigroups;

  * N: all finite nilpotent semigroups;

  * D: all finite semigroups in which the idempotents are
  right zeros;

  * LSl: all finite local semilattices.

In the whole paper, A denotes a finite alphabet.
Let V be a pseudovariety of semigroups. The free pro-V semigroup generated by A is denoted AV.
Its elements are pseudowords over V.
When V≠ I, as the associated  generating mapping A→ AV is
injective, one considers A to be contained in AV.
If φ A→ S is a generating mapping of a pro-V
semigroup, then we denote by φ_ V 
the unique continuous homomorphism AV→ S extending φ.

If V contains N, then the subsemigroup of
AV generated by A is isomorphic to A^+ and its elements are
the isolated points of AV, in view of which
A^+ is considered to be contained in AV, and the elements of A^+
and AV∖ A^+ are respectively called
the finite and infinite pseudowords over V.

By a topological semigroup, we mean a semigroup endowed with a
topology that makes the semigroup multiplication continuous. Unlike
some authors, we require that a compact space be Hausdorff. By a
compact semigroup, we mean a compact topological semigroup.
See <cit.>.

We denote by S^I the monoid obtained from the semigroup S by
adjoining to S an element denoted by 1 which acts as the identity.
Every semigroup homomorphism φ S→ T is extended to
a semigroup homomorphism S^I→ T^I, also denoted φ, such that φ(1)=1.
If S is a topological semigroup, then S^I
is viewed as a topological monoid whose topology is the sum
of the topological spaces S and {1}, whence 1 is an isolated
point of S^I.

We use the standard notation for Green's relations and its
quasi-orders on a semigroup S.
Hence, s≤_ Rt,
s≤_ Lt
and s≤_ Jt
respectively mean s∈ tS^I, s∈ S^It and s∈ S^ItS^I,
R, L, J 
are the associated equivalence relations, 𝒟=ℛ∨ℒ, and ℋ=ℛ∩ℒ.

A semigroup S has unambiguous ≤_ L-order if, for every
x,y,z∈ S,  x≤_ Ly and x≤_ Lz implies
y≤_ Lz or z≤_ Ly. One also has the dual notion of
unambiguous ≤_ R-order.  An unambiguous semigroup
is a semigroup with unambiguous ≤_ R-order and unambiguous
≤_ L-order.  The next proposition is an important tool to show
one of our main results.


  Let A be a finite alphabet.
  Let u,v∈ AA. Then u=v
  if and only if φ_ A(u)=φ_ A(v),
  for every mapping φ from A
  onto an unambiguous finite aperiodic semigroup.



  The “only if” direction of the statement is immediate.
  To establish the “if” direction, it suffices to show that it is
  the inverse limit of A-generated unambiguous finite aperiodic
  semigroups.

  It is well known that every A-generated finite aperiodic semigroup
  is a homomorphic image of an unambiguous A-generated finite
  aperiodic semigroup, namely its Birget-Rhodes expansion (also
  called iterated Rhodes expansion), cut down to the set of
  generators A <cit.>. Since pairs of distinct
  points of AA may be separated by continuous homomorphisms into
  finite aperiodic semigroups, the result follows.  






  


§ EQUIDIVISIBILITY  AND PSEUDOVARIETIES CLOSED UNDER CONCATENATION



A language L⊆ A^+ is said to be V-recognizable if
there is a homomorphism φ:A^+→ S into a semigroup S
from V such that L=φ^-1φ(L). We say that a
pseudovariety V of semigroups is closed under
  concatenation if, for every finite alphabet A, whenever L and
K are V-recognizable languages of A^+,
the set LK is
also a V-recognizable language of A^+.



  The following conditions are equivalent for a pseudovariety V of
  semigroups.
  
  
  *  V is closed under concatenation;
  
  *  A V= V;
  
  * 
    V contains N and the multiplication in AV is an
    open mapping for every finite alphabet A.
  


The
equivalence <ref>⇔<ref>
in Theorem <ref> is
from <cit.>. The difficult part of the
theorem is the equivalence
<ref>⇔<ref>,
which is a particular case of a more general result established by
Chaubard, Pin and Straubing <cit.>. The
latter, in turn, extends an earlier result of
Straubing <cit.>, establishing that a nontrivial
pseudovariety V of monoids satisfies A
V= V if and only if, for every finite alphabet A, whenever L
and K are V-recognizable languages of A^∗, the set LK
is also a V-recognizable language of A^∗. In the case of
semigroups, the absence in Theorem <ref> of reference to
the pseudovariety I of trivial semigroups is not surprising if
we take into account that A^+ is I-recognizable but not A^+
A^+, where we view these languages as languages of A^+.
 
Schützenberger <cit.> proved that a language
over a finite alphabet is A-recognizable if and only if it is
star-free, in the sense that it can be obtained from finite languages
by using only finite Boolean operations and concatenation. In
particular, it follows that A is closed under concatenation. As
important classes of examples of pseudovarieties closed under
concatenation that include A, one has the complexity
pseudovarieties C_n (cf. <cit.>)
and every
pseudovariety H formed by the finite semigroups whose
subgroups belong to the pseudovariety of groups H.

Combined with Theorem <ref>,
the next lemma, which will be quite useful in the sequel,
provides yet another characterization
of the pseudovarieties closed under concatenation.
A weaker version of the direct implication
was proved in <cit.>.


  Let S be a topological semigroup whose
  topology is defined by a  metric.
  The following conditions
  are equivalent:
  
  
  * 
    The multiplication in S is an open mapping;
  
  *  For every
    u,v∈ S, if (w_n)_n is a sequence of elements of S
    converging to uv, then there are sequences (u_n)_n and
    (v_n)_n of elements of S^I such that w_n=u_nv_n, lim
    u_n=u, and lim v_n=v.
  



  Consider a metric d inducing the topology of S.
  We denote by B(t,ε) the open
  ball in S with center t and radius ε.

  <ref>⇒<ref>:
  Let k be a positive integer.
  Since the  multiplication is an open mapping, the
  set B(u,1/k)
  B(v,1/k) is an open neighbourhood 
  of uv. Hence there is p_k such that
  w_n∈ B(u,1/k)
    B(v,1/k)
    if n≥ p_k.
  Let n_k be the strictly increasing sequence recursively defined by
  n_1=p_1 and  n_k=max{n_k-1+1,p_k} whenever k>1.
  Then there are sequences (u_n)_n and (v_n)_n satisfying the following
  conditions: if n_k≤ n<n_k+1 then
  u_n∈ B(u,1/k),
  v_n∈ B(v,1/k), and w_n=u_nv_n;
  and if n<n_1 then u_n=1 and v_n=w_n.
  The pair of sequences (u_n)_n and (v_n)_n satisfies
  condition <ref>.

  <ref>⇒<ref>:
  We want to prove that B(s,ε)B(t,ε) is open,
  for every s,t∈ S and ε>0. Let (w_n)_n be a
  sequence of elements of S converging to an element of
  B(s,ε)B(t,ε). Let u∈ B(s,ε) and
  v∈ B(t,ε) be such that lim w_n=uv. Take sequences
  (u_n)_n and (v_n)_n as in the statement of
  condition <ref>. There is
  N such that d(u_n,u)<ε-d(u,s) for all n≥ N. Then
  d(u_n,s)≤ d(u_n,u)+d(u,s)<ε for all n≥ N.
  Similarly, d(v_n,t)<ε for all sufficiently large n.
  Therefore, since w_n=u_nv_n, we have w_n∈
  B(s,ε)B(t,ε) for all sufficiently large n,
  which proves that B(s,ε)B(t,ε) is open.


A semigroup S is said to be
equidivisible <cit.> if,
for every equality of the form xy=uv, with x,y,u,v∈ S, there
exists t∈ S^I such that, either xt=u and y=tv, or x=ut and
ty=v. Clearly, free semigroups and groups are equidivisible.
Moreover, all completely simple semigroups are equidivisible.
Actually, a semigroup S is completely simple if and only if, for
every x,y,u,v∈ S such that xy=uv, there are t,s∈ S such that
xt=u, y=tv, x=us and sy=v <cit.>. Note
that every equidivisible semigroup is unambiguous. The converse is not
true: for instance, free bands are unambiguous, which follows easily
from the solution of the word problem for free bands (see, for
instance <cit.>) but not equidivisible for
more than one free generator since, if a,b are two distinct free
generators in a free band then, for x=a, y=b, u=v=ab, we have
xy=uv, yet y>_ℒv and x>_ℛu.
More generally, it is shown in <cit.> that, if V is a pseudovariety of
semigroups such that V=RB V, where RB
denotes the pseudovariety of finite rectangular bands, then AV
is unambiguous, for every finite alphabet A.

Let us say that a pseudovariety of semigroups V is
equidivisible if AV is equidivisible, for every finite
alphabet A. The following result was established
by the first two authors <cit.>, where 
denotes the Mal'cev product, LI the pseudovariety of all 
finite locally trivial semigroups, and CS the pseudovariety of
all finite completely simple semigroups.


  A pseudovariety of semigroups V is equidivisible if and only
  if  V=LI V or V⊆CS.


In particular, every
pseudovariety closed under concatenation is equidivisible. Many of our
results below are formulated not in terms of pseudovarieties but more
abstractly for free profinite semigroups with suitable properties, which
are satisfied for free profinite semigroups over pseudovarieties that are
closed under concatenation or, sometimes, more generally,
equidivisible.



§ THE QUASI-ORDER OF 2-FACTORIZATIONS



By a quasi-order on a set we mean a reflexive transitive
relation. In case the relation is additionally anti-symmetric, the
quasi-order is called a partial order. A quasi-ordering
(X,≤), in the sense of a set X with a quasi-order ≤, is
said to be total, or linear if x≤ y or y≤ x,
for every x,y∈ X.



 §.§ Definition and properties



Let S be a semigroup.
A 2-factorization of s∈ S is a pair (u,v) of elements of S^I
such that s=uv.
We denote the set of 2-factorizations of s by 𝔉(s).
We introduce in 𝔉(s)
a relation ≤ defined by (u,v)≤(u',v') if there exists t∈  S^I such
that ut=u' and v=tv', in which case we say that t
is a transition from (u,v) to (u',v').
The relation ≤ is a quasi-order. Concerning transitivity,
we have more precisely that if t is a transition from
(u,v) to (u',v')
and t' is a transition from (u',v') to (u”,v”),
then tt' is a transition from (u,v) to (u”,v”).

Given a quasi-order ≤ on a set P, we denote by ∼
the equivalence relation on P induced by ≤
and we write p<q if p≤ q but not p∼ q.
Denote
by ≺ the relation on P such that p≺ q
if and only if
q is a successor of p (equivalently, p is
a predecessor of q), that is,
p≺ q if and only if p<q
and
p≤ r≤ q ⇒ (r∼ p∨ r∼ q).

For every element s of a semigroup S, the quotient set
𝔉(s)/∼ is denoted 𝔏(s). We denote the
quotient mapping 𝔉(s)→𝔏(s) by χ. The
partial order on 𝔏(s) induced by the quasi-order ≤ on
𝔉(s) is also denoted by ≤. For
p,q∈𝔏(s), we also write p≺ q if p is a
predecessor of q. Sometimes we will also consider the unions
𝔉(S)=⋃_s∈ S𝔉(s) and
𝔏(S)=⋃_s∈ S𝔏(s).

The following result is immediate.


  
  A semigroup S is equidivisible if and only if 𝔏(s) is
  linearly ordered for every s∈ S.


The previous lemma
is the departing point
motivating this paper. For a good supporting reference on the theory
of linear orderings, see <cit.>.

We proceed to extract
from topological assumptions on S
some consequences on the quasi-order of
2-factorizations.
In what follows, 𝔉(s) is viewed as a topological
subspace of S^I× S^I.


  If S is a compact semigroup, then, for every
  s∈ S, the quasi-order ≤ on 𝔉(s)
  is a closed subset of 𝔉(s)×𝔉(s).



  Suppose (p_i,q_i)_i∈ I is a convergent net of elements of
  𝔉(s)×𝔉(s) with limit (p,q) and
  such that p_i≤ q_i for every i∈ I. Then, for each i∈ I,
  there is t_i∈ S^I making a transition from p_i to q_i. Since
  S^I is compact, the net (t_i)_i∈ I has a subnet converging
  to some t∈ S^I.
  Then, by continuity of multiplication on S^I, one deduces that
  indeed p≤ q, with t being a transition from p to q.



We shall denote the open intervals of a quasi-ordered
set P by

    ]←,p[={r∈ P: r<p},
        ]p,→[={r∈ P: p<r},
        ]p,q[=]p,→[∩]←,q[,

for every p,q∈ P. Considering the relation ≤, we also have
the intervals of the form ]←,p]={r∈ P: r≤
p}, [p,→[={r∈ P: p≤ r}, and so on.
Recall that the order topology of a linearly ordered set P is
the topology with subbase the sets of the form
]←,p[ and ]p,→[. In particular, we
consider the order topology on 𝔏(s).


  
  Let S be a compact equidivisible semigroup. For every s∈ S,
  the mapping χ𝔉(s)→𝔏(s) is
  continuous.



  It is sufficient to show that the sets of both forms
  χ^-1(]←,q[) and
  χ^-1(]q,→[) are open. By duality, we are
  actually reduced to show that χ^-1(]q,→[) is
  open. Since 𝔏(s) is linearly ordered by
  Lemma <ref>,
  the complement of this last set is
  χ^-1(]←,q]), which we therefore want to show
  to be closed. Consider a net (r_i)_i∈ I of elements of
  χ^-1(]←,q]), converging to some r∈𝔉(s). Let q̂∈χ^-1(q). Then r_i≤q̂
  for every i∈ I. It follows from
  Lemma <ref> that
  r≤q̂, that is, r∈χ^-1(]←,q]),
  showing that χ^-1(]←,q]) is closed.



  
  Let S be a compact equidivisible semigroup. Then, for every s∈ S,
  the order topology of 𝔏(s) is compact.
  Moreover, if the space S is metrizable, then
  the space 𝔏(s) is also metrizable and
  
  the set of isolated points of 𝔏(s) is countable.



  Since S is compact, 𝔉(s) is compact, being the
  preimage in S^I× S^I under multiplication of the closed set
  {s}, it is a closed subset of a compact space, whence compact.
  Since 𝔏(s) is clearly
  Hausdorff, it follows from Proposition <ref> that 𝔏(s) is compact.

  Suppose that S is metrizable. Then 𝔉(s) is
  metrizable, being a subspace of a product of two metrizable spaces.
  Since the continuous image of a compact metric space in a Hausdorff
  space is metrizable (<cit.>), it also follows
  from Proposition <ref> that 𝔏(s) is
  metrizable.
  As a compact metrizable space, 𝔏(s) has a dense
  countable subset.
  Since isolated points belong to every dense subset, they
  form a countable set.


Recall that a linearly ordered set L is said to be complete
if every subset of L which is bounded above has a least upper bound
(i.e., a supremum) or, equivalently, if every subset of L which is
bounded below has a greatest lower bound (i.e., an
infimum) <cit.>.
  

  
  Suppose S is a compact equidivisible semigroup.
  Then the linearly ordered set 𝔏(s) is
  complete.



  Let X be a subset of 𝔏(s). Consider the subset Y
  of 𝔏(s) of lower bounds of X and assume that it is
  nonempty.
  As S is equidivisible, we know by
  Lemma <ref> that the quasi-order ≤ on
  the set χ^-1(U) is linear, whence in particular this set is
  directed.
  Therefore, we may consider, in the product space
  S^I× S^I, the net (p)_p∈χ^-1(U). By compactness,
  this net has a subnet (p_i)_i∈ I converging to some element
  q of 𝔉(s).
  We claim that χ(q)=max U. If r∈χ^-1(X), then we have p≤ r for all p∈χ^-1(U),
  and so q≤ r, by
  Lemma <ref>,
  showing that q∈χ^-1(U). On the other hand, if p∈χ^-1(U), then, by the definition of subnet, there exists
  i_0∈ I such that p≤ p_i for all i≥ i_0. Hence we have
  p≤ q, again by
  Lemma <ref>. This
  proves the claim that χ(q)=max U, and so χ(q) is the
  infimum of X.




 §.§ The category of transitions



A directed graph with vertex set V and edge set E, which are
assumed to be disjoint, is given by mappings α,ω:E→ V
assigning to each edge s its source α(s) and
its target ω(s).
A semigroupoid is a directed graph, with a nonempty set of edges, endowed with
a partial associative binary operation on the set of its edges such that if s and t are edges, then
st is defined if and only if
ω(s)=α(t), in which case
α(st)=α(s) and ω(st)=ω(t).

Semigroupoids can be viewed as generalizations of semigroups, which in
turn can be viewed as one-vertex semigroupoids.
In particular, Green's relations generalize straightforwardly to Green's relations between the edges in semigroupoids.
For instance, in a semigroupoid S, s≤_ Jt means that the edge t is a factor of the edge s and s Jt means that s≤_ Jt and t≤_ Js.
A subsemigroupoid of the semigroupoid S is a subgraph T of S, with a nonempty set of edges, such that
s,t∈ T implies st∈ T whenever ω(s)=α(t).
Also, an ideal of a semigroupoid S
is a subsemigroupoid I of S such that for every t∈ I and every
  s∈ S, ω(s)=α(t) implies st∈ I, 
  and ω(t)=α(s) implies ts∈ I.


A category is a semigroupoid such that,
for each vertex v, there is a loop 1_v at v
satisfying 1_vs=s and t1_v=t for every
edge s starting in v and every edge t ending in v.
This coincides with the notion of small category from Category Theory, except that we compose in the opposite direction. In doing so, we are following a common convention in Semigroup Theory, see for example <cit.>.

If the sets of edges and vertices
of a semigroupoid are both endowed with compact topologies,
for which the semigroupoid operation and the mappings
α and ω are continuous,
then the semigroupoid is said to be compact.

Let S be an arbitrary semigroup.
To each s∈ S, we associate a category T(s),
the category of transitions for s, as follows:


  * the set of vertices of T(s) is 𝔉(s);

  * we have an edge
  (u,v,t,x,y) from (u,v) to (x,y), which we may denote
  (u,v)(x,y),
  if t is a transition
  from (u,v) to (x,y) (thus implying (u,v)≤ (x,y)); we say
  that t is the label of the edge;

  * multiplication of consecutive edges is done by multiplying
their labels, that is, the product of
(u_1,v_1)(u_2,v_2)
an
(u_2,v_2)(u_3,v_3)
is (u_1,v_1)(u_3,v_3).


Note that the sets of vertices of the strongly connected components of the
category T(s) are precisely the ∼-classes of 𝔉(s).

The category of transitions for S,
denoted T(S), is the coproduct category ⋃_s∈ S T(s).
We denote by Λ the faithful functor T(S)→ S^I
mapping each edge (u,v,t,x,y) to t. We say that Λ is
the labeling functor associated to T(S).
We remark that if S is a compact semigroup, then T(S) is a
compact category, with the vertex and edge sets
respectively endowed with the subspace topology of (S^I)^2 and of
(S^I)^5. Note that Λ is continuous.


Suppose that in 𝔏(s) we have p≤ q.
An element t∈ S^I will be called a
transition from p to q if t is a transition from an element
of p to an element of q, in which case we use
the notation pq.

For future reference, it is convenient to
register the following remark,
concerning the relationship between T(u) and T(uv).


  
  Let u,v be elements of a semigroup S.
  If (α,β)(γ,δ)
  is an edge of T(u),
  then (α,β v)(γ,δ v) is an edge
  of T(uv).


This remark is applied in the proof of the following lemma,
which in turn will later be used in the proof of Theorem <ref>.


  
  Let S be an equidivisible semigroup.
  Consider two edges σ and τ
  of T(S)
  with the same target
  and such that α(σ)<α(τ).
  Then the label of σ is a suffix of the label of τ.



  Let σ be the edge
  (α,β)(ε,φ) and
  τ be the edge
  (γ,δ)(ε,φ),
  with (α,β)<(γ,δ). The following equalities
  hold: ε=α t=γ z, β=tφ, and
  δ=zφ. From the equality α t=γ z and by
  equidivisibility, we deduce that if z is not a suffix of t, then
  there exists s∈ S such that γ s=α and st=z, that
  is, we have an edge (γ,z)(α,t) in T(ε). By Remark <ref>, there is an
  edge (γ,δ)(α,β) in T(S),
  which contradicts the hypothesis that
  (α,β)<(γ,δ). Hence z is a suffix
  of t.




§ THE MINIMUM IDEAL SEMIGROUPOID
  AND THE J-CLASS
  ASSOCIATED TO A ∼-CLASS


In a strongly connected compact semigroupoid C, there is an
underlying minimum ideal semigroupoid K(C) which may be
defined as follows. Consider any vertex v of C and the local
semigroup C_v of C at v, that is, the semigroup formed by the
loops at v. Then C_v is a compact semigroup, and therefore it has a minimum ideal K_v. Let K(C) be the subsemigroupoid of C with the same set of vertices of C and whose edges are those edges as C which admit some (and therefore
every) element of K_v as a factor. The next lemma is folklore.


  
  If C is a strongly connected compact semigroupoid, then K(C) is
  a closed ideal of C whose definition does not depend on the choice
  of v. Moreover, the edges
  in K(C) are J-equivalent, more precisely they are J-below every edge of C.


Let (u,v)∈𝔉(S).
An element z∈ S^I stabilizes (u,v) if
z labels a loop of T(S) at (u,v).
Note that the set M_(u,v) of stabilizers of
(u,v) is a monoid and that M_(u,v)
is the isomorphic image,
under the labeling functor Λ,
of the local
monoid of T(S) at (u,v).

Assume S is a compact semigroup. For p∈𝔏(S), let
T_p be the strongly connected component of T(S) whose
vertices are the elements of p. We denote by K_p the minimum
ideal semigroupoid K( T_p). Since ΛT(S)→
S^I is a (continuous) functor, where S^I is viewed as the set of
edges of single vertex semigroupoid, in view of
Lemma <ref> the set of labels of edges in K_p is contained in a single J-class of S^I, which we
denote J_p. For every (u,v)∈ p,
the minimum ideal of M_(u,v), which we denote 
I_(u,v), is the image under Λ of the minimum ideal
of the local monoid of T(S) at (u,v),
whence I_(u,v)⊆ J_p.
Note that J_p is regular, since I_(u,v) is itself regular.
The set J_p can also be characterized
as the set of J-minimum transitions from p to itself,
as seen in the next lemma.


  Let S be a compact semigroup, and let p∈𝔏(S).
  Then t is a transition from p to p
  if and only if t is a factor of the elements of J_p.



  Let (u,v) (x,y) be a transition between elements of
  p. Since (u,v)∼ (x,y), there is a transition
  (x,y) (u,v). The loop (u,v) (u,v)
  is a factor of every element ε in the minimum ideal of
  the local monoid at (u,v). Therefore, ts is a factor of Λ
  (ε)∈ J_p.

  Conversely, suppose that
  t is factor of the elements of J_p.
  Then there is a loop
  (u,v) (u,v)
  in 𝒦_p such that z=xty for some x,y∈ S^I.
  In 𝒯(S)
  we have the following path:
  (u,v)
  
  (ux,tyv)
  
  (uxt,yv)
  
  (u,v).
  Therefore,
  (ux,tyv)
  
  (uxt,yv)
  is an edge of 𝒯_p.




We next give some results that further highlight the role of
idempotent stabilizers of 2-factorizations of elements of S,
specially those idempotents in a J-class of the form J_p.

Recall that a semigroup is stable if
𝒥∩≤_ℒ=ℒ and
𝒥∩≤_ℛ=ℛ. In particular, any
compact semigroup is stable, see for
instance <cit.>.


  
  Let S be a stable unambiguous semigroup. Let e,f be idempotents stabilizing an element (u,v) of 𝔉(S). If e J f then
  e=f.



  The hypothesis gives ue=u=uf
  and ev=v=fv. Since S is unambiguous, from ue=uf we
  get e≤_ Lf or f≤_ Le. By
  stability, as e Jf, it follows that e
    Lf. Dually, from ev=fv we get e Rf. Hence e=f.





  Let S be a compact unambiguous semigroup,
  p∈𝔏(S) and (u,v),(x,y)∈ p.
  The edge (u,v) (x,y) of T(S)
  belongs to 𝒦_p if and only if
  t∈ J_p.



  The “only if” part holds by definition of J_p.
  Conversely, suppose that t∈ J_p.
  Denote by ε the edge
  (u,v) (x,y).
  As J_p is regular, there
  is an idempotent e∈ J_p such that t=et.
  Since t is a prefix of v, we have v=ev,
  thus we may consider the edges
  (u,v) (ue,v),
  (ue,v) (ue,v)
  and (ue,v) (x,y),
  respectively denoted by α, β and γ.
  Observe that ε=αβγ,
  and so it suffices to show that the loop β belongs
  to 𝒦_p.
  The ideal 𝒦_p contains
  the minimum ideal of the local monoid of T(S) at
  (ue,v).
  The latter contains an idempotent, of the form
  (ue,v)(ue,v) for some f∈ J_p.
  But f=e by Lemma <ref> and therefore ε∈𝒦_p.


Note that in the next lemma one does not assume that S is unambiguous.



  
  Let S be a compact semigroup, and let (u,v)∈𝔉(S).
  Let e be an idempotent stabilizing (u,v).
  If f is an idempotent J-equivalent
  to e, then f stabilizes an element
  of the ∼-class p of (u,v).
  Moreover, if e labels a loop
  of 𝒦_p, then
  f also labels a loop of 𝒦_p.



  If f Je, then there are in the J-class of e
  some elements s,t such that sts=s, tst=t, st=e, ts=f.
  We have the
  four edges in T(S) which are depicted in
  Figure <ref>.
  In particular, f
  stabilizes a vertex ∼-equivalent to (u,v).
  
  Denote by ε, σ, ϕ, τ the edges in
  Figure <ref> labeled by e, s, f, t,
  respectively. Since s=es, t=te, f=ts, we have
  σ=εσ, τ=τε and
  ϕ=τσ. Therefore, if ε belongs to the ideal
  𝒦_p, then all edges in Figure <ref> belong
  to 𝒦_p, and so f labels a loop of 𝒦_p.





  Let S be a compact semigroup, and let p∈𝔏(S).
  Every idempotent of J_p labels a loop of 𝒦_p.

For e∈ J_p, denote by p_e the nonempty set of elements of p stabilized by e.


  
    Let S be a compact unambiguous semigroup.
  Let p∈𝔏(S).
  Then J_p is the set of labels of edges of K_p.
  Moreover, if s∈ J_p and e and
  f are idempotents such that e Rs Lf,
  then s labels an edge from p_e to p_f. Moreover,
  there is a bijection p_e→ p_f,
  given by μ_s(u,v)=(us,tv),
  where t is the unique t∈ J_p such that st=e and ts=f.



  Let s be an element of J_p and let e and f be idempotents
  such that e Rs Lf. Then there exists (a
  unique) t∈ J_p such that st=e and ts=f, for which we have
  e Lt Rf.
  Let (u,v)∈ p_e. Note that such a pair (u,v) exists by
  Corollary <ref>. Therefore, we are in
  the same situation as in the proof of
  Lemma <ref>, with the four edges
  depicted in Figure <ref> belonging to 𝒦_p
  by
  Corollary <ref>.
  If there is another edge (u,v)(x,y) in K_p
  with (x,y)∈ p_f, then x=us and v=sy, thus y=fy=tsy=tv.
  Hence, there is for each vertex in p_e exactly one edge labeled s
  into a vertex of p_f. This defines the function μ_s
  p_e→ p_f such that μ_s(u,v)=(us,tv). Finally, note that
  μ_s and μ_t are mutually inverse.


We finish this section with
a couple of observations concerning aperiodic semigroups, starting with the
next lemma.


  
  Let S be a compact aperiodic semigroup.
  Let p∈𝔏(S).
  If (u,v),(x,y) are elements of p
  stabilized by the same idempotent e of J_p,
  then (u,v)=(x,y).



  Since (u,v)∼(x,y), there are t,z∈ S^I such that there
  are edges (u,v)(x,y) and (x,y)(u,v)
  in the category T(S). Then we also have edges
  as in the following picture:
  
    (50,16)(0,0)
        Nframe=n,Nadjust=w(uv)(5,8)(u,v)(xy)(45,8)(x,y)[curvedepth=5](uv,xy)ete[curvedepth=5](xy,uv)eze[loopangle=180](uv)e[loopangle=0](xy)e

  By Lemma <ref> and stability of S,
  we conclude that ete and eze are
  H-equivalent to e, thus, by aperiodicity, we get ete=eze=e.
  By the definition of the category T(S), we deduce that
  u=ue=x and v=ey=y.


In the following result,
we have a case in which the idempotents of
J_p parameterize the elements of p.


  
  Let S be a compact and unambiguous aperiodic semigroup.
  Let p∈𝔏(S). Then there is a bijection between the ∼-class p and
  the set of idempotents in J_p, sending each (u,v) to the unique idempotent e∈ J_p that stabilizes (u,v).



  Let (u,v)∈ p.
  There are in J_p idempotents that stabilize (u,v),
  as J_p contains the minimum ideal of the monoid of stabilizers of (u,v).
  If e,f are idempotents of J_p stabilizing (u,v), then e=f by
  Lemma <ref>.
  Hence, we can consider the function ε p→ J_p
  sending (u,v) to the unique idempotent
  of J_p stabilizing (u,v). The function ε
  is injective by Lemma <ref>,
  and it is surjective by Corollary <ref>.






§ FINITELY CANCELABLE SEMIGROUPS



Consider a compact semigroup S generated by a closed set A.
Recall that, in the context of topological semigroups, that means that
every element of S is arbitrarily close to products of elements of A.
Note that, since A is closed, we have S=S^IA=AS^I. Indeed, every
element of S is the limit of a net of the form (w_ia_i)_i∈ I,
where the a_i∈ A and the w_i are perhaps empty products of
elements of A. By compactness, we may assume that the nets
(w_i)_i∈ I and (a_i)_i∈ I converge in S^I, say to w
and a, respectively. Since A is closed, we conclude that a∈ A,
which shows that S⊆ S^IA.

Say that S is right finitely cancelable with respect to A
when, for every a,b∈ A and u,v∈ S^I, 
 the equality ua=vb implies
 a=b and u=v.
 This implies A∩ SA=A∩ AS=∅.
 

 Say that S is right finitely cancelable
 if it is finitely cancelable with respect to some closed generating subset A.
 It turns out that the set A is uniquely determined by S, as shown next.

 
   Let S be a compact semigroup generated by closed subsets A and B
   such that A∩ SA=B∩ SB=∅. Then we have A=B.   
   In particular, if S is right finitely cancelable with respect to A and
   to B, then A=B.
 

 
   Let a∈ A. Since S=S^IB=S^IA, we have a=sb for some
   s∈ S^I and b∈ B, and b=tc for some t∈ S^I and c∈ A.
   We obtain the factorization a=stc. Since A∩ SA=∅,
   we must have s=t=1, and so a=b∈ B, showing that
   A⊆ B. By symmetry, we have B⊆ A.
 
 
 

 
 Say that a pseudovariety of semigroups is right finitely
   cancelable if AV is right
 finitely cancelable with respect to A, for every finite alphabet
 A.
 

  
    A pseudovariety of semigroups V
    is right finitely cancelable if and only if V= D∗ V.
 

 
   It is observed in <cit.> that V is right
   finitely cancelable if and only if, for every finite alphabet A,
   and for every V-recognizable language L of A^+ and a∈
   A, the language La is also V-recognizable.
   In <cit.> one finds a proof that this is equivalent
   to V= D∗ V.
 

 The above definitions have obvious duals which are obtained by replacing
 right by left. Note that a semigroup pseudovariety V is right finitely cancelable if and only the pseudovariety
 V^op of semigroups of V with reversed multiplications is left finitely cancelable.
 We say that a compact semigroup is
 finitely cancelable (with respect to A)
 if it is simultaneously right and left
 finitely cancelable (with respect to A). Similarly, a pseudovariety of semigroups is finitely cancelable if it is simultaneously right and
 left finitely cancelable.
 


  If V is a semigroup pseudovariety containing some nontrivial
  monoid and such that V= V∗ D, then V is
  finitely cancelable (cf. <cit.>
  and <cit.>).


The following proposition is <cit.>.


  If V is an equidivisible pseudovariety of semigroups
  not contained in CS, then V
  is finitely cancelable.


The next lemma is the first of a series of results in which
the hypothesis of a semigroup being finitely cancelable enables us to
get further insight into the quasi-order of 2-factorizations.


  
  Suppose S is a compact semigroup, finitely cancelable with
  respect to A.
  Let u,v∈ S^I and a∈ A. If
  the ∼-class of at least one of (ua,v) and (u,av) is not a
  singleton, then (u,av)∼(ua,v).



  By duality, it suffices to consider
  the case where the ∼-class of p=(ua,v) is not a singleton.
  Let q be in p/∼ with p≠ q.
  As p≤ q, we may consider a transition x from p to q.
  Then we have q=(uax,y) for some y∈ S^I such that v=xy.
  Since q≤ p, there is t  such that ua=uaxt and y=txy.
  Because p≠ q, we must have t≠ 1,
  whence we may take b∈ A and z∈ S^I such that t=zb.
  Because S is finitely cancelable with respect to A,
  from ua=uaxt=uaxzb we get a=b and u=(ua)(xz).
  On the other hand, we have (xz)(av)=x(za)v=xtxy=xy=v,
  which shows that (ua,v)∼(u,av).


We now turn our attention to profinite semigroups.


  
  Suppose S is a profinite semigroup
  generated by a closed subset A.
  Let p,q∈𝔉(s) with p<q. Then,
  there are x,y∈ S^I and a∈ A such that 
  
    p≤(x,ay)<(xa,y)≤ q.
  



  Let p=(u,v) and q=(u',v').
  Since (u,v)<(u',v'), there exists t∈ S such that u'=ut
  and v=tv', and the system
  
    {[  utX  = u; Xtv' = v' ].

  has no solution X∈ S.
  By a standard compactness argument which can be found
  in the proof of <cit.>, there is some
   continuous onto homomorphism φ_0 S→ R,
   with R finite,
   which may be naturally extended to
   an onto continuous homomorphism φ S^I→ R^I,
   and  such that the following
  system (<ref>) has no solution
  X∈ R:
  
    {[  φ(u)φ(t)X     = φ(u); Xφ(t)φ(v')   = φ(v'). ].

  Let (t_n)_n be a net of elements of
  the (discrete) subsemigroup of S generated
  by A such that (t_n)_n converges to t and
  such that φ(t_n)=φ(t) for all n.
  Write t_n=a_n,0a_n,1⋯ a_n,k_n, with the a_n,i∈
  A. Then the following inequalities hold for i=0,…,k_n:
  
    (φ(ua_n,0⋯ a_n,i-1),φ(a_n,i⋯
        a_n,k_nv'))
        
           ≤
        (φ(ua_n,0⋯ a_n,i),φ(a_n,i+1⋯ a_n,k_nv')).

  Since ≤ is a transitive relation and
  the non-existence of a solution to 
  (<ref>) guarantees that the following
  strict inequality holds
  
    (φ(u),φ(a_n,0⋯ a_n,k_nv'))
      <
      (φ(ua_n,0⋯ a_n,k_n),φ(v')),

  we deduce that there is i=i_n such that the inequality
  (<ref>) is also strict.
  As A is closed and S is compact, by taking subnets we
  may assume that the net (a_n,i_n)_n
  converges to some a∈ A,
  that φ(a_n,i_n)=φ(a) for every n,
  and that each of the nets t'_n=a_n,0⋯
  a_n,i_n-1 and t”_n=a_n,i_n+1⋯ a_n,k_n converges to
  some t',t”∈ S^I, respectively
  (in particular, this yields t=t'at”). Then the strict inequality
  in (<ref>), with i=i_n, yields
  (φ(ut'),φ(at”v'))<(φ(ut'a),φ(t”v')),
  which implies that
  
    p=(u,t'at”v')
      ≤(ut',at”v')
      <(ut'a,t”v')
      ≤(ut'at”,v')=q.

  Thus, it suffices to choose x=ut' and y=t”v' to obtain the
  inequalities of the statement of the proposition.



We close this subsection with a result
regarding the existence of a successor in the quasi-ordered set
of 2-factorizations.


  Suppose S is a profinite semigroup, finitely cancelable with
  respect to A.
  Let p,q∈𝔉(s) and suppose that p<q.
  
  
  * 
    Consider the unique u,v,a such that
    u,v∈ S^I, a∈ A and p=(u,av).
    If p≺ q, then we have 
    p=(u,av)≺(ua,v)=q.
 
    Moreover, the ∼-classes of p and q are singletons.
  
  *  Conversely, if p=(u,av) and
    q=(ua,v), where u,v∈ S^I and a∈  A, then we have p≺ q.
    



  <ref>
  Notice that u,v,a really exist and are unique.
  Indeed, take p=(u,w), with u,w∈ S^I.
  One has w≠ 1, because p<q, and so w=av for some a∈ A and v∈ S^I, which are unique because S is finitely cancelable with respect to A.   
  By  Proposition <ref>,
  there are u',v'∈ S^I and a'∈  A
  such that p≤ (u',a'v')<(u'a',v')≤ q.
  Since p≺ q,
  we must have
  p∼ (u',a'v')<(u'a',v')∼ q.
  It then follows from Lemma <ref>
  that
  the ∼-classes of (u',a'v') and (u'a',v') are singletons,
  thus p=(u',a'v') and q=(u'a',v').
  By the uniqueness of u,v and a,
  we then have q=(ua,v)
  
  
  <ref> Assume there exists r=(x,y) with
  p<r<q. There are z,t such that x=ut, av=ty, ua=xz
  and y=zv.  If t=1, then r=p, while if z=1, then r=q,
  hence both t and z are different from 1. Since av=ty,
  from the fact that S is finitely cancelable with respect to A,
  it follows that there is t' such that t=at' and
  v=t'y.
  Similarly, there is z' such that z=z'a
  and u=xz'.
  Therefore, u=xz'=utz'=ua· t'z' and
  v=t'y=t'zv=t'z'· av. This shows that p∼ q, in contradiction
  with p<q. Hence p≺ q.




§ STEP POINTS AND STATIONARY POINTS



In this section, we continue gathering important
  properties of the linear orders induced by pseudowords. We identify
  two types of elements in such orders, that we call step points and
  stationary points. Let us start by introducing these
  notions.

Let P be a partially ordered set. We call
step points the points of P
that admit either a successor or a predecessor,
or are the minimum or the maximum of P, if they exist.
All other points are
said to be stationary. The set of step points of L
will be denoted by (L), and the set of stationary points of L
will be denoted by (L).

For an element s of a semigroup S, we also say that
p∈𝔉(s) is a step point
(respectively, a stationary point) if χ(p) is a step point
(respectively, a stationary point) of 𝔏(s).
In this section we further develop the results obtained in
Section <ref>, using the notions
of step point and stationary point.
If S is profinite and finitely cancelable, then the ∼-class
p of a step point
(u,v) of 𝔉(S)
is a singleton
(cf. Proposition <ref><ref>),
for which reason, in that case, we feel free to make the abuse of
notation p=(u,v).

As a preparation for the following example, recall that in
a compact semigroup S, if s∈ S, then s^ω
denotes
the unique idempotent in the closed subsemigroup of S generated by s. Later on, we
shall also make use of the notation s^ω+1 for s^ω s,
and s^ω-1 for the inverse of s^ω+1 in the maximal
subgroup containing s^ω+1.



  Consider the pseudoword a^ω of the free pro-aperiodic
  semigroup {a}A.
  Then 𝔉(a^ω) has only one stationary point, namely
  (a^ω,a^ω).
  The set 𝔉(a^ω)
  is linearly ordered, whence isomorphic to 𝔏(a^ω),
  with order type ω+1+ω^∗.
  More precisely, its elements are ordered as follows:
  
    (1,a^ω)<(a,a^ω)<(a^2,a^ω)
    <⋯ <(a^ω,a^ω)<⋯
    <(a^ω,a^2)<(a^ω,a)<(a^ω,1).




Example <ref> should be compared with the
following one.


  Consider the pseudoword a^ω of the free profinite semigroup
  AS.
  Like in Example <ref>,
  𝔏(a^ω) has
  order type
  ω+1+ω^*,
  and its sole stationary point is p=(a^ω,a^ω)/∼:
  
    (1,a^ω)<(a,a^ω-1)<(a^2,a^ω-2)
          <⋯ <p<⋯
          <(a^ω-2,a^2)<(a^ω-1,a)<(a^ω,1).

    
  But (a^ω,a^ω)/∼ has infinitely many elements,
  namely, the pairs of the form (g,g^ω-1), where g
  is an element in the maximal subgroup containing a^ω.


Examples <ref>
and <ref> fit in the following definition.

[Clustered sets]
  We say that the linearly ordered set P is
clustered if the following conditions hold:


  *  P has a minimum min P and a maximum
  max P;

  *  for every q∈ P, if q=min P
  or q has a predecessor,
  then q has a successor or q=max P;

  *  for every q∈ P,
  if q=max P or q has a successor,
  then q has a predecessor  or q=min P;

  * 
  for every p,q∈ P, if ]p,q[ is nonempty, then
  there is a step point in the interval ]p,q[.




Property <ref> translates
into saying that the set of step points of P is dense
with respect to the order topology of P.




  Let S be a profinite semigroup which is finitely cancelable,
  and let s∈ S. Then 𝔏(s) is clustered.



  Property <ref> in Definition <ref>
  holds trivially, with min𝔏(s)=(1,s)
  and max𝔏(s)=(s,1).

  Let us show <ref>.
  Take s∈ S and p,q∈𝔉(s) with p≺ q.
  We have to show that either q has a successor, or q=max P.
  Let A be the generating set with respect to
  which S is finitely cancelable.
  By Proposition <ref>, there are u,v∈ S^I
  and a∈ A with p=(u,av) and q=(ua,v).
  If v=1, then q=max P, so that we may assume v≠1.
  Let v=bw with b∈ A, and let
  r=(uab,w).  Clearly, we have q≤ r. We claim that q≺
  r. Indeed, if q=r, then ua=uab and bw=w. Therefore, a=b,
  u=ua, and av=aaw=aw=v, showing that p=q, in contradiction
  with the hypothesis. Hence, we must have q<r, since otherwise we
  would obtain q∼ r and q≠ r, which entails p∼ q by
  Lemma <ref>.
  Finally, by Proposition <ref><ref> applied to q
  and r, we get q≺ r.
  This establishes <ref>, and <ref>
  holds dually.

  Finally, let us prove <ref>.
  If p is a step point,
  then ]p,q[≠∅
  implies that the successor of p belongs to ]p,q[. Hence,
  it suffices to consider the case where p is stationary.
  Let p̂∈χ^-1(p)
  and q̂∈χ^-1(q).
  By Proposition <ref>, there are
  u,v∈ S^I and a∈ A such that
  p̂≤(u,av)<(ua,v)≤q̂
  in 𝔉(s).
  By Proposition <ref><ref>, we
  have (u,av)≺(ua,v).
  Therefore, (u,av) and (ua,v) are step points.
  In particular, we have (u,av)∈]p,q[∩ (𝔏(s)).



In the next result, we characterize the stationary points as the vertices
with a nontrivial local monoid in the category of transitions.



  
  Let S be a profinite semigroup
  which is finitely cancelable. Let (u,v)∈𝔉(S).
  Then (u,v) is a stationary point if and only if
  it is stabilized by some element of S.





  Suppose that (u,v) is stationary.
  Let A be the set with respect to which S is finitely cancelable.
  Since v≠ 1, we may take a factorization v=aw with a∈ A and w∈ S^I.
  Clearly,  (u,aw)≤ (ua,w) holds, and so (u,aw)∼ (ua,w) by
  Proposition <ref><ref>.
  Hence, there is an edge (ua,w) (u,aw), for some
  t∈ S^I. This implies that at∈ S stabilizes (u,v).
  
  Conversely, suppose there is z∈ S
  stabilizing  (u,v). There is some factorization of
  the form z=at, for some a∈ A and t∈ S^I. The following
  equalities and inequalities
  
    (u,v)=(u,atv)≤(ua,tv)≤(uat,v)=(u,v)

  show that (u,v)=(u,atv)∼(ua,tv).
  It follows from
  Proposition <ref><ref> that (u,v) has no successor,
  since otherwise this successor would be (ua,tv), in
    contradiction with (ua,tv)≤(u,v). Since it is not the maximum
  of 𝔉(s) (because v=zv≠ 1), it must be a stationary point.


Proposition <ref>
is applied in the proof of the next result.


  
   Let S be an equidivisible profinite semigroup S
     which is finitely cancelable.
     Then, in T(S), any two coterminal edges between
       elements of distinct strongly connected components  are
       equal. In other words, if
     (α,β)(γ,δ)
     and (α,β)(γ,δ)
     are edges of
     T(S) such that (α,β)<(γ,δ), then t=s.
 


  The hypothesis translates into
  the following equalities: γ=α t=α s and
  β=tδ=sδ. We first assume that at least one of the
  points (α,β) and (γ,δ) is a step point. By
  symmetry, we may as well assume that (α,β) is a step
  point. From the equality α t=α s, by equidivisibility
  and without loss of generality, we may assume that there is some
  z∈ S^I such that α z=α and t=zs.
  Then, we have zβ=zsδ=tδ=β.
  This shows that
  (α,β)(α,β)
  is a loop of T(w). Since (α,β)
  is assumed to be a step point,
  applying Proposition <ref>
  we obtain z=1, thus t=s.

  It remains to consider the case where both (α,β) and
  (γ,δ) are stationary. By
  Theorem <ref>, there is a step point
  (x,y)∈𝔉(w) such that
  (α,β)<(x,y)<(γ,δ). By the preceding paragraph,
  there are unique edges in T(w) from (α,β) to
  (x,y) and from (x,y) to (γ,δ). Let r_1 and r_2 be
  the respective labels. To prove the proposition, it suffices to show that
  τ=r_1r_2 whenever (α,β)(γ,δ) is an
  edge of T(w).

  Note that
  (x,r_2) and (α,τ) are elements of 𝔉(γ).
  Since β=τδ and y=r_2δ, if (x,r_2)≤ (α,τ) (in 𝔉(γ)),
  then (x,y)≤(α,β) (in 𝔉(w)) by
  Remark <ref>, which gives a contradiction.
  Hence, by equidivisibility, there is in T(γ) an edge (α,τ)(x,r_2).
    In particular τ=r_0r_2.
 Remark <ref> guarantees the
 existence of the edge
 (α,β)(x,y). But we defined, in the previous paragraph, the edge (α,β)(x,y) as the unique edge from (α,β) to (x,y).
 Therefore r_0=r_1, and so we have the equality τ=r_1r_2, which we have seen
 to be sufficient to conclude the proof.



  
        Let S be an equidivisible profinite semigroup S
     which is finitely cancelable.
     Let p_1,p_2∈𝔏(w).
     If p_1<p_2 then the set of transitions from p_1 to p_2 is
     contained in a J-class of S.



      Let x_1,x_1'∈ p_1 and x_2,x_2'∈ p_2,
      and consider edges x_1x_2
      and x_1'x_2'.
      Then we have edges x_ix_i'
      and x_i'x_i,
      and also
      x_1x_2
      and
      x_1'x_2'.
      By Proposition <ref>,
      we must have t=s_1t'r_2
      and t'=r_1ts_2, whence t Jt'.





  We remark that there is a large class of pseudovarieties
  whose corresponding finitely generated relatively free profinite semigroups satisfy the hypotheses of Proposition <ref>.
  Let V be a pseudovariety
  of semigroups such that V=LI V.
  Then every semigroup of the form AV,
  with A a finite alphabet,
  satisfies all conditions in Proposition <ref>:
  they are profinite, equidivisible,
  and finitely cancelable
  (cf. Theorem <ref>
  and Proposition <ref>, where the latter
  may be applied because
  V=LI V implies V⊇LI
  and thus V⊈CS).



Proposition <ref> is used in the proof of the
following lemma, establishing a sufficient condition for equality
between stationary points.


  
  Let S be an equidivisible profinite semigroup which is finitely
  cancelable.
  Let p and q be stationary points of 𝔏(S).
  If there is a transition pq such that t
  lies J-above both J_p and J_q then p=q.



  We have p≤ q, so, arguing by contradiction, suppose that p<q.
  Consider an edge
  (u,v)(x,y)
  of T(S) with (u,v)∈ p and (x,y)∈ q. Let e∈ J_p and
  f∈ J_q be
  idempotents respectively
  stabilizing (u,v) and (x,y). Then we have a transition
  (u,v)(x,y).
  By
  Proposition <ref>,
  it follows that etf=t,
  whence, since S is stable,
  and by the hypothesis that t lies J-above both e and f,
  we have e Rt Lf.
  Therefore, there exists s∈ J_p=J_q such that ts=e and st=f.
  Now, the assumption that we have a transition
  (u,v)(x,y)
  means that the equalities ut=x and v=ty hold. Combining
  with the equalities ue=u and fy=y, we deduce that xs=uts=ue=u
  and sv=sty=fy=y. Hence (x,y)≤(u,v), which
  contradicts the assumption p<q. To avoid the contradiction, we
  must have p=q.


The following is an example obtained by application of Lemma <ref>.


   Let V be a pseudovariety
   containing LSl, and let A be a finite alphabet.
   It is shown in <cit.> (see
   also <cit.>),
   using Zorn's Lemma and a standard compactness
   argument, that AV contains regular elements that
   are J-maximal among the elements of AV∖ A^+.
   We next verify that if V is closed under concatenation
   and u is a J-maximal regular element of
   AV, then the order type of 𝔏(u) is ω+1+ω^*.
   Indeed, let p,q be two stationary points
  of 𝔏(u) such that p≤ q,
  and let e∈ J_p, f∈ J_q.
  From the maximality assumption on u,
  we deduce that e Ju Jf.
  Hence p=q by Lemma <ref>, showing that
  𝔏(u) has only one stationary point.
   


Restricting our attention to the case of profinite semigroups
that are free relatively to pseudovarieties closed under concatenation,
we obtain stronger results about step and stationary
points. The next lemma is a provisional result with this flavor, which
is improved later on, namely in
Proposition <ref>.


  
  Let A be a finite alphabet, and let
  V be a pseudovariety closed under concatenation.
  Let w∈ AV.
  If (u,v) is a stationary point
  of 𝔉(w), then
  there exists a strictly increasing sequence (u_n,v_n)_n
  of points of 𝔉(w) such that lim u_n=u and lim v_n=v.



  Since A is finite,
  AV is metrizable,
  and so 𝔏(w)
  is metrizable by Corollary <ref>.
    Let p=(u,v)/∼.
  By  Theorem <ref>, there is a strictly
  increasing sequence (x_n,y_n)_n of step points
  converging in
  𝔏(w) to p.
  As S is compact, taking subsequences
  we may assume that
  (x_n,y_n)_n converges
  in 𝔉(w) to some (x,y)∈𝔉(w). By
  Proposition <ref>, we know that (x,y)∈ p.
  Then there are edges (x,y)(u,v) and (u,v)(x,y)
  in T(w). Note that lim x_n=x=xst. Hence, by
  Theorem <ref> and
  Lemma <ref>,
  for every n there is a factorization x_n=x_n's_nt_n
  with lim x_n'=x, lim s_n=s, and lim t_n=t. Then, we have
  lim(x_n's_n,t_ny_n)=(u,v) in 𝔉(w),
  thus lim(x_n's_n,t_ny_n)/∼=p in 𝔏(w) by
  Proposition <ref>.
  And since (x_n's_n,t_ny_n)≤(x_n,y_n)<(x,y),
  we conclude that the sequence  ((x_n's_n,t_ny_n)/∼)_n has p
  as a supremum but not as a maximum, enabling us to extract from the
  sequence (x_n's_n,t_ny_n)_n a strictly increasing subsequence.
  Since this subsequence also
  converges to (u,v) in 𝔉(w), this completes the proof.




  
    Let A be a finite alphabet, and let
    V be a pseudovariety closed under concatenation.
  Let w∈ AV and let K be a nonempty clopen subset
  of 𝔉(w). Then K has a minimum and a maximum, and both
  of them are step points.



  As K is closed, whence compact, we know from
  Proposition <ref> that its projection under
  χ is also compact,
  whence closed. Let p=infχ(K) and q=supχ(K). Since
  χ(K) is closed, we have p=minχ(K) and q=maxχ(K).
  Hence, there exist p̂∈ K∩χ^-1(p) and q̂∈
  K∩χ^-1(q). Although these points are perhaps not unique,
  all choices are ∼-equivalent. Moreover, for every other point
  r∈ K, from χ(r)∈[p,q] we deduce that p̂≤ r≤q̂.

  It remains to show that p̂ and q̂ are step points. Suppose on
  the contrary that p̂ is a stationary point. By
  Lemma <ref>, there is a strictly
  increasing sequence (p̂_n)_n in 𝔉(w) converging
  to p̂. Since K is open, it must contain points of the form
  p̂_n. As p̂_n<p̂, this contradicts the fact that
  p̂ is
  a minimum of K. Hence, p̂ is a step point and, similarly, so
  is q̂.


Lemma <ref> was used to prove
Lemma <ref>.
We next use
Lemma <ref>
to show that the sequence in Lemma <ref>
may be formed only by step points.


  
    Let A be a finite alphabet, and let
  V be a pseudovariety closed under concatenation.
  Let w∈ AV.
  If (u,v) is a stationary point
  of 𝔉(w), then
  there exists a strictly increasing sequence (u_n,v_n)_n
  of step points of 𝔉(w) such that lim u_n=u and
  lim v_n=v.



  Since
  𝔉(w) is a zero-dimensional space
  and the set of step points of 𝔉(w) is linearly
  ordered by ≤, it suffices to show that, for every clopen subset
  K⊆𝔉(w) containing p=(u,v)/∼ and every interval
  [q,p], with q<p, there is some step point in the intersection
  K∩[q,p].
  Because (𝔏(w)) is
  topologically dense in 𝔏(w)
  (cf. Theorem <ref>),
  we are reduced to the case where q is a step point.
  Let r be a step point such that r>p, for instance
  r=(w,1).
  The closed intervals in 𝔏(w) whose extremities
  are step points are clopen, hence, by Proposition <ref>,
  the interval [q,r] in 𝔉(w) is clopen.
  Therefore, the subset L=K∩[q,r] is also clopen.
    Note that L is nonempty, as p∈ L.
  Lemma <ref> guarantees that L
  has a minimum s which is a step point.
  It then follows from p∈ L that s∈ K∩[q,p],
  thereby completing the proof of the lemma.




§ CLUSTER WORDS



In this section, we use
  the knowledge obtained about the labeled linear orders induced by
  pseudowords over A to obtain a representation theorem: every
  pseudoword over A may be represented by a (partially) labeled
  linear order having specific properties, which we introduce now.

By a partially labeled ordered set, we mean a pair (P,f) such
that P is an ordered set and f is a function (the labeling)
with domain contained in P. An isomorphism between partially labeled
ordered sets (P,f) and (Q,g) is an isomorphism φ P→
Q of ordered sets such that g=φ( f) and
f(p)=g∘φ(p) for every p∈ f.

Consider a profinite equidivisible semigroup S which is finitely
cancelable with respect to A.
We may then consider the mappings  and  from S^I to
A⊎{1}
such that (1)= (1)=1 and, for s∈ S, the images (s)
and (s) are respectively the unique prefix and the unique
suffix of s in A.
For s∈ S, denote by 𝔏_c(s) the
partially labeled linearly ordered set (𝔏(s),ℓ) defined by the mapping
ℓ (L)→ A⊎{1} such that ℓ (u,v)=(v),
for (u,v)∈ (𝔏(s)).
Note that ℓ (p)=1 if and only if p=(s,1).
Recall from Theorem <ref> that 𝔏(s)
is clustered. By a cluster word over A
we mean a partially labeled linearly ordered set (L,ℓ) such that L is
clustered and ℓ is a function (L)→ A⊎{1}.


   For the pseudoword w=(a^ω b)^ω of {a,b}A, the
    cluster word
  𝔏_c(w) is
  described in Diagram (<ref>),
    
    aa⋯_ω ∙ ⋯ aa_ω^*b
            aa⋯_ω ∙ ⋯
              aa_ω^*baa⋯_ω ∙ ⋯ aa_ω^*b
            ⋯_ω
           ∙⋯aa⋯_ω ∙ ⋯ aa_ω^*b
            aa⋯_ω ∙ ⋯
              aa_ω^*b aa⋯_ω ∙ ⋯ aa_ω^*b _ω^*

  where each of the small bullets
  ∙ represent a stationary point q such that
  a^ω∈ J_q and
  the bigger bullet
  ∙
  represents the unique stationary point r such that
  (a^ω b a^ω)^ω∈ J_r.
  Note that, in particular, the
  order type of 𝔏(w) is
  (ω+1+ω^*)ω+1+(ω+1+ω^*)ω^*.



We next introduce, in Definition <ref>, a notion of
algebraic recognition of cluster words inspired by the definition of
automata recognizing words indexed by linear orders, introduced
in <cit.>. A notorious similarity resides in the
role played by cofinal sets, whose definition we next recall.
In a linearly ordered set L, a subset X of L is left
  cofinal at p if X∩]q,p[≠∅ for every q<p,
right cofinal at p if X∩]p,q[≠∅ for
every q>p, and cofinal if it is right or left cofinal at p.


    
    Let φ A→ S be a generating mapping of a
    semigroup S,  and let s∈ S.
    We say that the cluster word (L,ℓ) over A
    is recognized by the pair (φ,s)
    if there is a mapping g(L)→𝔉(s)
    satisfying:


  * g(min L)=(1,s);

  * g(max L)=(s,1);

  * if p_1≺ p_2 in L, then
  g(p_1)g(p_2)
  is an edge of T(s);

  * if p is a stationary point of L
  then, for every q∈𝔉(s),
  the set g^-1(q) is left cofinal at p if and only if
  it is right cofinal at p.

If such conditions are satisfied,
then we say that
(L,ℓ) is g-recognized by the pair (φ,s).
We also say that
(L,ℓ) is recognized by the pair (φ,s)
if it is g-recognized, for some g.
Finally, when (L,ℓ) is g-recognized by (φ,s), we define
F_g to be the function
from (L) to the power set 𝒫(𝔉(s))
  such that
  
    F_g(p)   ={q∈𝔉(s):g^-1(q) is left cofinal at
          p}
       ={q∈𝔉(s):g^-1(q) is right cofinal at
          p},

  for every stationary point p of L (cf. <ref>).





  
If f (L',ℓ')→ (L,ℓ) is an isomorphism
of cluster words over A and if (L,ℓ) is g-recognized by (φ,s),
then (L',ℓ') is (g∘ f)-recognized by (φ,s).
Hence, the property of
(L,ℓ) being recognized by (φ,s) is invariant under
isomorphism of cluster words.


The next lemma, concerning the function F_g, will be
applied in Section <ref>.


  
 Consider a cluster word (L,ℓ) over A,
 recognized by the pair (φ,s), where
 φ A→ S is a generating mapping of a finite semigroup S
 and s∈ S.
 Suppose that the order topology of L is metrizable.
 Let p be a stationary point of L.
 Then the following properties hold:
 
 
  * if P is a subset of 𝔉(s)
  such that F_g^-1(P) is cofinal at p, then P⊆
  F_g(p);

  * there is an open interval I containing p such that
  F_g(q)⊆ F_g(p) for every stationary point q
  in I.
 



  <ref>
  Without loss of generality, suppose that
  F_g^-1(P) is left cofinal at p.
  Then, taking into account that L is metrizable,
  there is a strictly increasing sequence (p_n)_n of
  stationary points of L converging to p
  such that F_g(p_n)=P for all n.
  Consider a metric d on L
  inducing the order topology of L. 
  For each k≥ 1, let ]q_k,p[ be an open interval of
  L such that d(q_k,p)<1/k
  Then, there is 
  n_k such that p_n_k∈]q_k,p[.
  Let (s_1,s_2)∈ P.
  By the definition of F_g, we know that 
  g^-1(s_1,s_2) is left cofinal at p_n_k. Therefore, there is
  in the interval ]q_k,p_n_k[ a step point belonging to g^-1(s_1,s_2). Since
  ]q_k,p_n_k[⊆]q_k,p[, this proves that
  g^-1(s_1,s_2) is left cofinal at p, thus
  (s_1,s_2)∈ F_g(p). We have therefore proved that the inclusion
  P⊆ F_g(p) holds.

  <ref> Let P be the set of subsets P
  of 𝔉(s) such that F_g^-1(P) is not cofinal at
  p.  For each P∈ P, there is an open interval I_p
  containing p such that F_g^-1(P)∩ I_p=∅.
  Because S is a finite semigroup,
  the set P is finite, and so I=⋂_P∈ PI_p is an open interval of L containing p.  Let q be a stationary point in I.  Let
  R=F_g(q).  Suppose R∈ P. Then q∈ I_R. But we also have
  q∈ F_g^-1(R), which leads to
  F_g^-1(R)∩ I_R≠∅, a contradiction with the
  definition of I_R.  This shows that R∉ P.  We then
  deduce from part <ref> of the lemma that
  R⊆ F_g(p).


The main results in this section
are about cluster words defined by elements of AA,
but in the next proposition we embrace
without additional effort all pseudovarieties closed under concatenation.


  
  Consider a pseudovariety V closed under
  concatenation.
Given w∈ AV, and a generating mapping
φ A→ S of a semigroup S of V,
let s=φ_ V(w).
Consider the mapping
g_w,φ(𝔏(w))→𝔉(s)
such that
g_w,φ(u,v)=(φ_ V(u),φ_ V(v))
for every 
step point (u,v) of 𝔏(w).
  Then the cluster word 𝔏_c(w)
  is g_w,φ-recognized by (φ,s).



  The conditions <ref>-<ref>
  in Definition <ref>
  for g_w,φ-recognition by  (φ,s) are
  clearly satisfied, and <ref>
  follows directly from
  Proposition <ref><ref>.

  Let us verify condition <ref>.
    Let (x,y)∈𝔉(s) be such that
    g_w,φ^-1(x,y) is left
  cofinal at p. Then, there is a strictly increasing sequence
  (u_n,v_n)_n≥ 1 of step points
  belonging to g_w,φ^-1(x,y)
  converging in 𝔉(w) to an element (u,v) of p.
    Since (φ_ V(u_n),φ_ V(v_n))=(x,y) for every n≥ 1,
  we also have (φ_ V(u),φ_ V(v))=(x,y),
  by continuity of φ_ V.
  By the dual of Proposition <ref>,
  there is a strictly decreasing sequence
  (u'_n,v'_n)_n≥ 1 of step points converging to (u,v) in
  𝔉(w).
  In particular, by continuity of φ_ V,
  for all sufficiently large n, we have
  (φ_ V(u_n'), φ_ V(v_n')) = (x,y),
  and so g_w,φ^-1(x,y) is right cofinal at p.
  Dually, if
  g_w,φ^-1(x,y) is right cofinal at p then
  g_w,φ^-1(x,y) is left cofinal at p. This establishes 
  condition <ref>.


In the case of unambiguous aperiodic semigroups,
we have a converse for
Proposition <ref>, as stated
in the next theorem.


    Let w∈ AA,
    and consider
    a generating mapping  φ A→ S of a finite aperiodic
  unambiguous semigroup S.
  Then φ_ A(w)=s if and only if
  𝔏_c(w) is recognized by (φ,s).


We defer the proof of
Theorem <ref>
to Section <ref> (but note that
the direct implication in
  Theorem <ref>
  is an immediate application of
  Proposition <ref>).
  Meanwhile, we use it to prove the following main result.



  Let u,v∈ AA. Then
  u=v if and only if the cluster words 𝔏_c(u)
  and 𝔏_c(v) are isomorphic.



  The isomorphism of cluster words is clearly necessary to have
  u=v. Conversely, suppose 𝔏_c(u) and
  𝔏_c(v) are isomorphic.
  Let φ A→ S be a generating mapping of
  a finite unambiguous aperiodic semigroup S. Take s=φ_  A(u).
  Then 𝔏_c(u) is recognized by
  (φ,s), according to the direct implication in
  Theorem <ref>.
  But then
  𝔏_c(v) is also recognized by
  (φ,s) (cf. Remark <ref>).
  Hence, we have s=φ_  A(v) by the converse implication
  in Theorem <ref>.
  By Proposition <ref>, this establishes u=v.





§ STABILIZERS


Given a semigroup S and s∈ S, we say that an element x
of S^I stabilizes s on the right if sx=s; the set
(s) of all such x constitutes a submonoid of S^I and is
called the right stabilizer of s. One defines dually the
elements that stabilize s on the left, which form a submonoid
(s) of S^I, called the left stabilizer of s.

An application of the following result will be required in the sequel. 


  
  Let S be an equidivisible profinite semigroup which
  is finitely cancelable.
  Let u∈ S.
  Let g be an element of the monoid (u)
  or of the monoid  (u).
  If g is regular within that monoid, then g=g^2.


In Theorem <ref>, the hypothesis that S is finitely
cancelable is not superfluous: if G is a group, then the semigroup
G^0 obtained from G by adjoining a zero is equidivisible, and
G^0 is the left and right stabilizer of zero.

We should mention that we do not know of any other examples of
equidivisible profinite semigroups that are finitely cancelable other
than free pro-V semigroups, where V is a pseudovariety with
suitable closure properties. For such semigroups,
Theorem <ref> follows from more general results
in <cit.>.
Nevertheless, since our results apply to all equidivisible profinite
semigroups that are finitely cancelable, we present a proof of
Theorem <ref> which may be of independent interest. 

As a first step we have the following simple
statement.


  
  Let x and y be D-equivalent
  elements of a stable semigroup S.
  If yx=x then y=y^2.



  Since yx=x and y are in the same D-class and S is
  stable, we must have y Rx. Therefore, y=xu for
  some u∈ S^I, so y^2=yxu=xu=y.


We proceed with an auxiliary lemma. 


  
  Let S be an equidivisible profinite semigroup
  which is finitely cancelable.
  Let g,w∈ S be such that gw=w. If (x,y)∈𝔉(w) is
  a step point  satisfying (g^ω,w)≤(x,y), then gx=x.



  Note that (g^ω,w)≤(x,y) implies
  x≤_ R g^ω, thus x=g^ω x, a fact that we shall
  use along the proof.

  By equidivisibility,
  (x,y) and (gx,y) are comparable in 𝔉(w).
  Suppose first that (gx,y)≤(x,y). Then there is
  t∈ S^I such that gxt=x and ty=y.
  It suffices to show that t=1. The equality gxt=x
  entails x=g^nxt^n for every n≥1, thus x=g^ω
  xt^ω=xt^ω.
  And since ty=y,
  we conclude that t^ω stabilizes (x,y)
  in T(S).
  Because (x,y)
  is a step point,  Proposition <ref>
  implies that t=1.

  Suppose next that (x,y)≤(gx,y). Then
  (g^ω-1x,y)≤(g^ω x,y)=(x,y)
  (cf. Remark  <ref>)
  and, by the preceding case, we deduce that g^ω-1x=x.
  With a left multiplication  by g on both sides of the latter equality, we
  obtain x=g^ω x=gx, as desired.




  It suffices to consider the case where g is an element of
  (u), as the other case is dual.

  We first establish the theorem when g is a group element, that is, g=g^ω+1.
  
  Let R be the set of elements (α,β)
  of 𝔉(u) such that
  α Rg.
  Note that R is nonempty, indeed (g,u)∈ R.   
  The set R is closed, whence compact, and so by continuity of
  χ,
  the image χ(R) is also compact, whence closed.
  Therefore, by completeness of 𝔏(u)
  (cf. Proposition <ref>),
  the closed set χ(R) has a maximum p=(x,y)/∼.
  Let us observe that, since
  two ∼-equivalent elements
  must have R-equivalent first components,
  the inclusion χ^-1(p)⊆ R holds. 
  Moreover, since g=g^ω+1,
  we have (g^ω,u)∈ R, thus (g^ω,u)≤ (x,y)
  by definition of p.
  
  Suppose first that p is a step point.  
  Then gx=x
  by Lemma <ref>. As x Rg,
  we conclude that g=g^2 by Lemma <ref>.

  If p is stationary, then,
  by Theorem <ref>,
  there is a
  net (x_i,y_i)_i∈ I
  of step points converging in 𝔏(u) to p
  and such that p<(x_i,y_i) for all i∈ I.
  By compactness, taking a subnet, we may assume that
  (x_i,y_i)_i∈ I
  converges in 𝔉(u) to
  some element (x',y') of p.
    As (g^ω,u)< (x_i,y_i),
    we deduce from Lemma <ref>
    that gx_i=x_i for every i∈ I.
   Taking limits, it follows that gx'=x'.
   Since  (x',y')∈ p, we have  x Rx', whence
   g Rx', and we again deduce that g=g^2 by
   Lemma <ref>.

   We have thus concluded the proof for the case where g is a group
   element of S. Let us now suppose
   that  g is regular within (u). Then there is
    h∈(u)
   such that g=ghg and h=hgh. Since hu=gu,
   by equidivisibility we know that h and g
   are ℛ-comparable in S. We actually have
   h Rg in S, by stability of S.
   Let z∈ S^I be such that h=gz. Then g=g^2zg,
   and therefore g Rg^2.
   This shows that g is a group element of S, and so,
   as we are in the case already proved, we get g=g^2.   



A semigroupoid S is trivial
if, for any two vertices p,q∈ S, there is at most one edge p→ q.


  
      Let S be an equidivisible profinite semigroup
      which is finitely cancelable.
      For every p∈𝔏(S),
      the ideal K_p is a trivial category.



  Let (u,v)(x,y) and
  (u,v)(x,y) be edges of
  𝒦_p.
  The proof amounts to showing that s=t.
  There is an edge (x,y)(u,v) in 𝒦_p.
  In particular,
  sz and (sz)^2
  label loops in 𝒦_p at vertex (u,v).
  Hence, sz and (sz)^2 belong to J_p,
  and so sz is a group element of S stabilizing u on the right.
  We then deduce from Theorem <ref> that sz
  is an idempotent of J_p, denoted by e, which
  stabilizes (u,v).
  Similarly, tz is an idempotent of J_p
  stabilizing (u,v). It follows
  from Lemma <ref>
  that sz=tz=e.
  Symmetrically, we have zs=zt=f, with f an idempotent of J_p.
  As s,t,z belong to J_p, this shows that s=t.







For a compact semigroup S
and p∈𝔏(S), let U_p
be the union of the maximal subgroups of J_p.
The next proposition, whose proof relies on
Theorem <ref>, should be compared with
Proposition <ref>.
We show
that U_p parameterizes in a natural way the class p,
without assuming aperiodicity, but assuming equidivisibility and
finite cancelability.



  
  Let S be an equidivisible profinite semigroup which is
  finitely cancelable.
  Let
  p∈𝔏(S) be a stationary
  point.
  Then we have a bijection ν_p U_p→ p, defined as follows:
  
  
  * for each idempotent e∈ J_p, fix an element 
  (u_e,v_e) of p stabilized by e;
  
  * to each g in the maximal subgroup H_e of S containing
    e, associate the element ν_p(g)=(u_eg,g^ω-1v_e) of p.
   



  We first verify that the function ν_p is well defined.
  Corollary <ref>
  guarantees that every idempotent e∈ J_p stabilizes
  some (u_e,v_e)∈ p.
  If g∈ H_e, then
  (u_eg,g^ω-1v_e)
  (u_e,v_e)
  and
    (u_e,v_e)
  (u_eg,g^ω-1v_e)
  are edges of T(S), whence
  (u_eg,g^ω-1v_e)∼ (u_e,v_e).
  It follows that ν_p is indeed well defined.


  We next show that ν_p is injective.
  If g∈ U_p then g^ω is an idempotent of J_p stabilizing
  ν_p(g)=(u_g^ωg,g^ω-1v_g^ω).
  Hence, by
  Lemma <ref>, if
  ν_p(g)=ν_p(h), then g^ω=h^ω,
  thus g,h
  belong to the same maximal subgroup
  and u_g^ωg=u_g^ωh.
  The latter is equivalent to
  u_g^ω=u_g^ωhg^ω-1,
  and so, by
  Theorem <ref>, we have hg^ω-1=g^ω.
  This means that g=h,
  thereby showing that ν_p is injective.

  It remains to show that ν_p is surjective. Let (u,v) be an
  arbitrary element of p and let e be an idempotent in J_p
  stabilizing (u,v). Since p is the set of vertices of a
  strongly connected component of T(S), there is some edge
  (u_e,v_e)(u,v), and whence also an edge
  (u_e,v_e)(u,v).
  By Lemma <ref>, it
  follows that ete∈ H_e. Hence, we must have ν_p(ete)=(u,v).


Note that, under the conditions of
Proposition <ref>,
the set p_e of elements of p stabilized by e is precisely
ν_p(H_e).



§ A CHARACTERIZATION OF THE J-CLASS
  ASSOCIATED TO A ∼-CLASS



Let S be an equidivisible compact semigroup, w∈ S and
p∈𝔏(w).
We define a subset L_p of S^I, depending only on p,
as follows.
Take an arbitrary strictly increasing sequence
p_1<p_2<⋯ 
converging to p
in 𝔏(w) — if such a sequence does not exist, for instance,
if p has a predecessor, then take L_p=∅.
For each m≥1 and n> m,
let t_m,n be a transition from p_m to p_n.
For fixed m≥ 1, let t_m be an accumulation point
of the sequence (t_m,n)_n> m.
If t is an accumulation point of the sequence (t_m)_m
then t∈ L_p, and every element of L_p is obtained by this
process, the sequence
p_1<p_2<⋯ being allowed to change.
Dually, taking strictly decreasing sequences converging to p, we
define a subset R_p of S associated with p.


  
  Let S be an equidivisible compact semigroup.
  For every p∈𝔏(S), the
  sets L_p and R_p are contained in J_p.



  Let w∈ S be such that p∈𝔏(w), and
  suppose that (p_n)_n is a strictly increasing sequence of
  elements of 𝔏(w) converging to p.
  For each m≥ 1 and n>m, let t_m,n be a transition from p_m
  to p_n, and let t_m be an accumulation point of
  the sequence (t_m,n)_n. Finally, let t be an accumulation point of the
  sequence (t_m)_m. The proof that L_p is contained in J_p is concluded once we show that t∈ J_p.

  We first claim that, for every fixed m≥1, t_m is a transition
  from p_m to p. For each n>m, choose (u_n,v_n)∈ p_m and
  (x_n,y_n)∈ p_n such that u_nt_m,n=x_n and v_n=t_m,ny_n.
  By compactness,
  the sequence (t_m,n,u_n,v_n,y_n)_n
  has some subnet
  (t_m,n_k,u_n_k,v_n_k,y_n_k)_n_k
  such that t_m=lim_k t_m,n_k and the nets (u_n_k)_k,
  (v_n_k)_k, and (y_n_k)_k converge, respectively to some u, v,
  and y.
  Since
  (u_n_k,t_m,n_ky_n_k)=(u_n_k,v_n_k)∈ p_m
  and
  (u_n_kt_m,n_k,y_n_k)=(x_n_k,y_n_k)∈ p_n_k, it follows
  from Proposition <ref>
  that (u,v)=(u,t_my)∈ p_m and (ut_m,y)∈ p, which proves the claim.

  Since (p_m)_m converges to p and t is an accumulation point of the
  sequence (t_m)_m, it follows
  again from Proposition <ref>
  that t is a transition from p
  to p.

  Choose z∈ J_p
  such that there is a loop (ut_m,y)(ut_m,y).
  Since (u,v)<(ut_m,y) and (u,v)(ut_m,y)
  is an edge of T(S), Lemma <ref>
  yields that z is a suffix of t_m. This holds
  for every m≥1, whence z is a suffix of t. As we
  have already shown that t is a transition from p to p, we
  deduce that t∈ J_p by Lemma <ref>.
  Hence we have L_p⊆ J_p and
  dually R_p⊆ J_p.




  
    Let S be an equidivisible profinite semigroup which is
  finitely cancelable.
  Let w∈ S. Suppose that (u_n,v_n)_n is a strictly
  increasing sequence in 𝔉(w) converging to a stationary
  point (u,v). For each pair m<n, let t_m,n∈ S be a
  transition (u_m,v_m)→(u_n,v_n).
  Then, for each m,
  the sequence (t_m,n)_n converges to
  the unique transition from
  (u_m,v_m) to (u,v).
  Moreover,
  the sequence (t_m)_m converges to the label of the only loop at the
  vertex (u,v) in the trivial category K_p, where p=(u,v)/∼.



  Every accumulation point of the sequence
  (t_m,n)_n labels an edge from
  (u_m,v_m) to (u,v). But there is only one such edge, by
  Proposition <ref>. Since
  we are dealing with a compact space, this implies
  that (t_m,n)_n converges to some element t_m.
  To conclude the proof, it suffices to
  show that every accumulation point t of the
  sequence (t_m)_m labels the same loop at vertex (u,v).
  By the definition of L_p, we have t∈ L_p,
  whence t∈ J_p by Theorem <ref>.
  Moreover, in T(w) the sequence
  of edges
  (u_m,v_m)(u,v)
  admits the loop
  (u,v)(u,v)
  as an accumulation point.
  Since t∈ J_p, this loop belongs
  to K_p
  by Corollary <ref>.
  By Corollary <ref>, there is only
  one loop of K_p at (u,v). Therefore,
  every accumulation point of (t_m)_m is the label of
  that loop.


In some cases, Theorem <ref> can be strengthened, as
seen next.


  
    Let A be a finite alphabet, and let
  V be a pseudovariety closed under concatenation.
  Then L_p=J_p=R_p for every stationary point p∈𝔏( AV).



  According to Theorem <ref>,
  it remains to prove
  that J_p is contained in L_p and in R_p.
  By symmetry, it suffices to
  prove that J_p⊆ L_p.

  Let τ be an element of J_p. Then by
  Proposition <ref>
  there are elements (u,v)
  and (u',v') of p such that (u,v)(u',v') is an
  edge of K_p. According to
  Proposition <ref>, there are
  strictly increasing sequences (q_n)_n and (q'_n)_n of step
  points of 𝔉(w) converging to (u,v) and (u',v'),
  respectively. We may define recursively
  a strictly ascending sequence of step points p_n as follows:
  
    
  * p_1=q_1;
    
  * if n>1 is even, then p_n
      is the smallest term of the sequence (q'_n)_n
      belonging to ]p_n-1,p[;         
    
  * if n>1 is odd, then p_n
      is the smallest term of the sequence (q_n)_n belonging to ]p_n-1,p[;      
  
  For each m≥1
  and each n≥ m
  let t_m,n
  be the unique transition from p_2m-1 to p_2n.
  Let t_m be an accumulation point of the sequence
  (t_m,n)_n≥ m.
  Since (p_2n)_n converges to
  (u',v'), the pseudoword t_m labels an edge
  from p_2m-1 to (u',v').
  Let t be an accumulation point of the sequence (t_m)_m.
  Since (p_2m-1)_m converges to
  (u,v), the pseudoword t labels an edge
  from (u,v) to (u',v').
  By the definition of L_p, we have t∈ L_p.
  Therefore
  t∈ J_p by Theorem <ref>.
  By Corollary <ref>,
  the category K_p is trivial,
  whence t=τ,
  thus proving that τ∈ L_p.






§ PROOF OF THEOREM <REF>



Throughout this section, when not explicitly stated, we consider w
to be an element of AA, where A is a finite alphabet, and
φ A→ S to be a generating mapping of a finite
aperiodic semigroup S. We also take s=φ_ A(w).

Consider a mapping g(𝔏(w))→𝔉(s).
From hereon, we assume that 𝔏_c(w) is g-recognized by the
pair (φ,s).  In particular, all properties of
  Definition <ref> are fulfilled when
  (L,ℓ)=𝔏_c(w).

[g-projection]
Let x and y be two step points of 𝔏(w)
such that x≤ y.
By Propositions <ref> (in case x<y)
and <ref> (in case x=y),
there is a unique edge
in T(w)
from x to y. Let t be its label, which is 1 if x=y.
If g(x) g(y) is
an edge of T(s),
then we say that the edge xy is g-projected (to
g(x) g(y)).


The unique edge from x to y will sometimes be denoted simply
by xy, without reference to the label.  



  
  Let x, y and z be step points such that
  x≤ y≤ z. If
  x y
  and y z
  are g-projected, then x z
  is g-projected.


  Given two step points x and y, write x≺≺ y
if x≤ y and the interval
[x,y] is finite. 



  
  If x≺≺ y, then x y is g-projected.



Let ≈ be the equivalence relation on (𝔏(w)) generated by
≺≺,
that is, x≈ y if and only
x≺≺ y
or y≺≺ x. The ≈-class of x is denoted [x]_≈.
Note that w∈ A^+
if and only if (1,w)≈ (w,1).
Let O_w be a subset of
(𝔏(w)) such that each
≈-class contains exactly one element of 
O_w, with  the additional restriction
that if w∉ A^+ then we have

    O_w∩ [(1,w)]_≈={(1,w)}  and   O_w∩ [(w,1)]_≈={(w,1)}.


[Bridges]
  A bridge in 𝔏(w), with respect to
  the mapping g,
  is a nonempty open interval I of 𝔏(w)
  such that, for every pair of step points
  x,y of I, with x<y,
  the edge x→y is g-projected.
  A special bridge in 𝔏(w)
  is a bridge of the form [X,Y[,
    with X,Y∈ O_w such that X<Y.


Notice that every nonempty interval contained in a bridge is also a bridge, and that special bridges are clopen intervals.



  Let F be a nonempty family of special bridges
  of 𝔏(w). 
  If ⋃ F is a closed interval,
   then ⋃ F is a special bridge.


Before proving Lemma <ref>, we remark that
the hypothesis that the union ⋃ F is closed cannot be removed.
Indeed, consider a case where we have a strictly increasing sequence (q_n)_n of stationary points converging to a stationary point q
(cf. Example <ref>).
Between q_n and q_n+1 pick an element p_n of O_w.
Then the union of the special bridges
[min𝔏(w),p_n[ is [min𝔏(w),q[,
which is not a special bridge.


  Since ⋃ F is a compact set having an open cover by the
  elements of F,
    we have ⋃ F=⋃ F'
    for some finite subfamily F' of F.
    Then, for some n≥ 1, we may assume that
    there are elements  X_1<X_2<⋯ <X_n and
    Y_1<Y_2<⋯ <Y_n of O_w
    such that
    
    F'={[X_k,Y_k[:1≤ k≤ n}.

    Consider the set
    
    Z={X_i:1≤ k≤ n}∪{Y_i:1≤ k≤ n}.

    and let Z_1<Z_2<⋯ <Z_m be the elements of Z. Notice that m≥ 2.
    For each k∈{1,…,m-1}, denote
    by I_k the interval [Z_k,Z_k+1[.
    In case Z_k=X_j for some j,
    we have Z_k+1≤ Y_j, and so I_k is a special bridge contained
    in the special bridge [X_j,Y_j[.
    If Z_k=Y_j for some j,
    then, since Z_k<Z_m, we must have Z_k∈⋃ F, whence
    Z_k∈[X_i,Y_i[ for some i.
    As Z_k<Y_i, we have Z_k+1≤ Y_i, and so  
    I_k is a special bridge contained
    in the special bridge [X_i,Y_i[.
    Hence,  the set
    
    F”={I_k:1≤ k≤ m-1}

    is a family of special bridges.
    
    Let U_k=⋃_j=1^k I_j.
    Note that U_k=[Z_1,Z_k+1[.
    We prove by induction on k∈{1,…,m-1} that
    U_k is a special bridge.
    The initial step is trivial. Suppose
    that U_k is a special bridge,
    for some k∈{1,…,m-2}.
    To prove that U_k+1 is
    a special bridge, consider two step points x,y in U_k+1 with x<y. We have to show that the edge x y is g-projected. Using
    induction, it suffices to consider the case where  x∈ U_k and y∈ I_k+1. Let Z'_k+1 be the
    predecessor of the step point Z_k+1 in 𝔏(w).
    Since U_k=[Z_1,Z_k+1[ and I_k+1=[Z_k+1,Z_k+2[,
    the edges x Z'_k+1 and Z_k+1 y
    are g-projected, and the same is true obviously for the edge
    Z'_k+1 Z_k+1;
    hence, by Remark <ref>,
  xy is g-projected. This proves that
    U_k+1 is a special bridge, concluding the induction.
    
    The result now follows because U_m-1=⋃ F”=⋃ F'=⋃ F.
  


  In the proof of the next lemma and in the sequel,
  we use the notation λ(x)=inf [x]_≈ and
     ρ(x)=sup[x]_≈, for a step point x.
     Note that λ(x) is stationary unless
     (1,w)≺≺ x, and ρ(x) is stationary unless
     x≺≺ (w,1). Also, when r∈(𝔏(w)),
the unique element of O_w∩ [r]_≈
is denoted O_w(r).

     
  
    Let X,Y∈ O_w be such that X<Y. Suppose that for every
    stationary point p in [X,Y[ there is a special bridge
    containing p. Then [X,Y[ is a special bridge.
  

  
    Let P=[X,Y[∩(𝔏(w)).
    By hypothesis, for each p∈ P
    there is a special bridge I_p containing p. Note that
    I_p∩[X,Y[ is also a special bridge containing p. Therefore, we may as well assume that I_p⊆[X,Y[. Let U=⋃_p∈ PI_p.
  Then U⊆[X,Y[.

  We claim that U=[X,Y[.
    As we trivially have P⊆ U,
    we are reduced to showing that,
    for every z∈[X,Y[∩(𝔏(w)),
    we have z∈ U.
    
    If ρ(X)<z<λ (Y) holds, then
    the stationary points ρ(z) and λ(z) belong to [X,Y[.
    Either z∈]λ(z), O_w(z)[ or z∈[
    O_w(z),ρ(z)[.
    As ]λ(z), O_w(z)[⊆ I_λ(z)
    and [ O_w(z),ρ(z)[⊆ I_ρ(z),
    it follows that z∈ U.

    Suppose that ρ(X)≥ z.
    Then z∈[X,ρ(X)[, thus z∈ I_ρ (X).
    Similarly, if λ (Y)≤ z  then
    z∈ I_λ (Y). In both cases z∈ U.

    We proved that U=[X,Y[. By
    Lemma <ref>, the interval [X,Y[
    is a special bridge.
  


  

  Let p_1 and p_2 be two elements of 𝔏(w)
  such that p_1≤ p_2.
  If p_1<p_2, then, as written in Corollary <ref>,
  the set of transitions from p_1 to p_2 is contained
  in a J-class J_p_1,p_2. In case p_1=p_2,
  let J_p_1,p_2=J_p_1.
  By a J-minimum transition from p_1 to p_2,
  we mean a transition from p_1
  to p_2 belonging to J_p_1,p_2; this terminology is useful to
  unify both cases p_1<p_2 and p_1=p_2 in some of our arguments.



  
    Let p_1,p_2∈𝔏(w), with p_1≤ p_2.
    For every (x_1,y_1)∈ p_1 and (x_2,y_2)∈ p_2,
    the intersection I of J_p_1,p_2
    with the set of labels of edges from (x_1,y_1) to
    (x_2,y_2) is a singleton. If
    t is the unique element of I,
    and if e_i is the unique idempotent of J_p_i
    that stabilizes (x_i,y_i) (cf. Lemma <ref>), then t=e_1te_2.
  


  
    If p_1<p_2, then the lemma follows
    straightforwardly from Proposition <ref>.
    If p_1=p_2=p, then 
    J_p_1,p_2=J_p.
    Since edges in 𝒯_p labeled by elements of
    J_p are edges of K_p
    by Corollary <ref>,
    and since K_p is trivial by
    Corollary <ref>, 
    we have also in this case that there is only one
    edge from (x_1,y_1) to (x_2,y_2)
    with label in J_p_1,p_2.
    Let e_i and t be as in the statement of the lemma.
    Because e_i stabilizes (x_i,y_i),
    we may consider the edge
    (x_1,y_1)(x_1,y_2), 
    and so t Je_1te_2.
    By the already proved uniqueness, we must have
    t=e_1te_2.
  

  [J-bridge]
    A pair of elements (p_1,p_2) of (𝔏(w)) is a
    J-bridge with respect to g, if
    
    
  * p_1≤ p_2;
    
  * F_g(p_1)∩ F_g(p_2)≠∅;
    
  * the elements of φ_ A(J_p_1)
      are J-equivalent to the elements of φ_ A(J_p_2);
    
  * if τ is a J-minimum
      transition from an element
      of p_1 to an element of p_2, then φ_ A(τ)
      is J-equivalent to the elements of φ_ A(J_p_1).    
    


Note that (p,p) is a J-bridge, whenever p is a stationary point.

 In the proof of the following
 proposition, the hypothesis that S is an unambiguous aperiodic semigroup
 is essential.
 

  
  Suppose that S is unambiguous and let (p_1,p_2) be a J-bridge with respect to g. Suppose there are step points z_1 and z_2 such that
  [z_1,p_1[ and ]p_2,z_2]
  are bridges with respect to g.
  Let (s_1,s_2)∈ F_g(p_1)∩ F_g(p_2).  
  Then there are step points x_1∈ [z_1,p_1[
  and x_2∈]p_2,z_2]
  such that
  x_1 x_2
  is g-projected to an idempotent loop
  of T(s) of the form (s_1,s_2) (s_1,s_2).



  Let I_1=[z_1,p_1[ and I_2=]p_2,z_2].
  By the definition of F_g(p_i), there are
  strictly monotone sequences (r_i,m)_m≥ 1 (increasing if
  i=1, decreasing if i=2)  of step points in I_i∩ g^-1(s_1,s_2),
  converging in 𝔉(w) to elements r_i of p_i (i=1,2).
    Denote by e_i
    the unique idempotent of J_p_i
    that stabilizes r_i.

    For each n≥ m, let t_1,m,n be the label of the unique
    edge from r_1,m to r_1,n,
    and let
    t_2,m,n be the label of the unique
    edge from r_2,n to r_2,m.
    By Corollary <ref>
    and its dual, for each i∈{1,2} the sequence (t_i,m,n)_n
    converges to an element t_i,m, and
    in turn the sequence (t_i,m)_m converges to e_i.
    Therefore, by continuity of φ_ A,
    for each i∈{1,2} we may
    take m_i≥ 1 and n_i≥ m_i for which we have
    
    φ_ A(e_i)=φ_ A(t_i,m_i)=φ_ A(t_i,m_i,n_i).


   Our hypothesis yields that the edge
  r_1,m_1 r_1,n_1
  is g-projected
  to the edge
  (s_1,s_2)(s_1,s_2).
  In particular, (s_1,s_2)
  is stabilized by φ_ A(e_1). Similarly,
  (s_1,s_2)
  is stabilized by φ_ A(e_2).
  Therefore, φ_ A(e_1)=φ_ A(e_2), by
  Lemma <ref>,
  since the finite semigroup S is unambiguous.

  By Lemma <ref>,
   the unique minimum J-transition τ
    from r_1 to r_2 is such that
    τ=e_1τ e_2.
  By the definition
  of J-bridge, we
  have φ_ A(e_1) Jφ_ A(τ).
  On the other hand, τ≤_ Re_1 and τ≤_ Le_2.
  Hence φ_ A(e_1) Hφ_ A(τ) because
  φ_ A(e_1)=φ_ A(e_2) and S is stable. Therefore,
  
    φ_ A(e_1)=φ_ A(τ)=φ_ A(e_2),

  since S is aperiodic.       

  The unique edge (cf. Proposition <ref>)
  from r_1,m_1 to r_2,m_2,
  with label ζ,
  is factorized by the edges
  r_1,m_1 r_1,
  r_1 r_2
  and
  r_2 r_2,m_2.
  Therefore ζ=t_1,m_1τ t_2,m_2.
  From (<ref>)
  and (<ref>) we get
  φ_ A(ζ)=φ_ A(e_1).
  Therefore, r_1,m_1 r_2,m_2
  is g-projected to the idempotent loop
  (s_1,s_2) (s_1,s_2).
  Since r_i,m_i is a step point in I_i, this concludes the proof.


We next show a pair of lemmas where the function g does not appear,
but which will be used later in this section in the proof of
Theorem <ref>.


  
  Consider a pseudovariety V closed under concatenation.
  Let w be an element of AV.
  Consider a generating mapping φ A→ S of a semigroup in V.
  For every stationary point p of 𝔏(w),
  there is a clopen interval I containing p
  such that, if t is a transition between elements q,r of I
  with q≤ r, then
  φ_ V(t) is
  a factor of the elements of φ_ V(J_p).



  Let (u,v)∈ p. By
  Proposition <ref>
  and its dual, there is in 𝔉(w) an increasing sequence
  (p_n)_n
  of step points
  converging to (u,v)
  and a decreasing sequence
  (p'_n)_n
  of step points converging to (u,v).
  Denote by τ_n
  the unique edge
  from p_n to (u,v),
  and by
  τ'_n
  the unique edge
  from (u,v) to p'_n (uniqueness of these edges is
  from Proposition <ref>).
  Let ε be a loop in T_p at vertex (u,v).
  Then τ_nετ'_n converges to a loop ε̃ of
  T(w) at vertex (u,v). Since
  ε is a factor of ε̃,
  we must have Λ(ε̃)∈ J_p
  (thus we have ε̃=ε by
  Corollary <ref>, although we
  shall not use that fact).
  Therefore, since φ_ A and the labeling mapping Λ
  are continuous,
  there is N such that
  φ_ V(Λ(τ_nετ'_n))∈φ_ V(J_p)
  for all n≥ N.
  Consider the clopen interval I=[p_N,p'_N].
  Let q and r be elements of I such that q≤ r,
  and let t be a transition from q to r. Then
  t is a factor of a transition from p_N to p'_N.
  But Λ(τ_Nετ'_N) is the unique transition
  from p_N to p'_N, by Proposition <ref>. Hence,
  φ_ V(t) is a factor of the elements of φ_ V(J_p).



  Let (q_n)_n be a sequence of stationary points of
  𝔏(w)  converging
  to q. For each n, let u_n be an element of
  J_q_n. Suppose that (u_n)_n converges to u.
  Then u is J-above J_q.



  The sequence of edges
  (q_nq_n)_n of T(w)
  converges to
  qq.
  Since u labels a transition from q to itself,
  u is J-above J_q.


In Lemma <ref> it is not true
in general that u∈ J_q.
In Example <ref>,
the stationary point r such that (a^ω ba^ω)^ω
belongs to J_r is the limit of a sequence (q_n)_n of stationary points 
such that a^ω∈ J_q_n, but a^ω∉ J_r.

[Mapping Γ]
  Let J_S be the set of regular J-classes of S, and
  denote by Υ the mapping
  (𝔏(w))→ J_S sending a stationary
  point p to the J-class containing φ_ A(J_p).
  Denote by Γ the mapping
  Υ× F_g(𝔏(w))→ J_S×Im(F_g)
  sending p to (Υ(p),F_g(p)).


In the next lemma, we continue to assume that 𝔏_c(w)
is g-recognized by (φ,s).


  Suppose that the semigroup S is unambiguous.
  Let X,Y be elements of O_w such that X<Y.
  Suppose that the restriction of Γ to
  (𝔏(w))∩[X,Y[ is constant.
  Then [X,Y[ is a special bridge.



  Let p∈(𝔏(w))∩[X,Y[.
  By Lemma <ref>,
  it suffices to prove that there is a special bridge
  containing p.
  Let I be a clopen interval containing p
  satisfying the properties described in
  Lemma <ref>,
  and let [x_0,y_0]
  be the clopen interval I∩[X,Y[, which is indeed nonempty as it
  contains p.
  Note that x_0 and y_0 are step points.
  Let X'= O_w(x_0)
  and Y'= O_w(y_0).
  By the definition of O_w, x_0≥ X implies
  X'≥ X, and y_0≤ Y implies
  Y'≤ Y, thus [X',Y'[⊆[X,Y[.
  Therefore, we have p∈[X',Y'[ and
  [ρ(X'),λ(Y')]⊆ I∩[X,Y[.
  
  Let x,y be step points in [X',Y'[ such that
  x<y.
  We want to show that x y is g-projected.
  If
  x≺≺ y,  then we apply
  Remark <ref>,
  so we suppose that is not the case,
  which implies ρ (x)≤λ(y).
  Note that [ρ(x),λ(y)]⊆[ρ(X'),λ(Y')].
  If t is a J-minimum
  transition from
  ρ (x) to λ(y)
  then t≤_ JJ_ρ(x)
  and t≤_ J J_λ(y) by Lemma <ref>,
  thus
  φ_ A(t)≤_ JΥ(ρ(x))
  and φ_ A(t)≤_ JΥ(λ(y)).
  Hence
  φ_ A(t)≤_ JΥ(p),
  because the restriction of Υ
  to (𝔏(w))∩[X,Y[ is constant.
  By the choice of I
    (cf. Lemma <ref>),
    this implies φ_ A(t)∈Υ(p).
  Therefore, since the restriction of Γ
  to (𝔏(w))∩[X,Y[ is constant,
  we have shown that the pair (ρ (x),λ(y)) is a J-bridge.
     By Remark <ref>,
     [x,ρ(x)[
     and ]λ(y),y]
     are bridges, whence it follows from
     Proposition <ref>
     that there are step points x'∈[x,ρ(x)[
     and y'∈]λ(y),y] such that
     x' y' is g-projected.
     Therefore x y is g-projected
     by Remarks <ref>
     and <ref>.     
     This concludes the proof that [X',Y'[ is a special bridge
     containing p.



  Suppose that the finite aperiodic semigroup S is unambiguous.
  Then every stationary point p
  of 𝔏(w) is contained in some special bridge.



  Endow J_S×Im(F_g) with the following partial order:

    (J_1,P_1)≤ (J_2,P_2) (J_2<_ JJ_1 or
      (J_1=J_2 and P_1⊆ P_2)).
  
For each p∈(𝔏(w)), let
  
    G(p)={Γ(q) : q∈(𝔏(w)) and Γ(q)<Γ(p)}.

  We shall prove by induction on the cardinal of G(p)
  that p is contained in some special bridge.
  
  We start with some preliminary remarks.  
  Let I=[α,β] be a clopen interval containing p
  satisfying the properties described in
  Lemmas <ref>
  and <ref><ref>.
  Let X= O_w(α) and Y= O_w(β).
  Then the following holds:
  
    (𝔏(w))∩ I=
      (𝔏(w))∩[X,Y[.

  Let q∈(𝔏(w))∩[X,Y[.
  Then we have Υ(p)≤_ JΥ(q)
  by the choice
  of I
  (cf. Lemma <ref>).
  It also follows from the choice of I
  (cf. Lemma <ref><ref>)
  that F_g(q)⊆ F_g(p).
  Hence we have
  
    Γ(q)≤Γ(p)  for all q∈(𝔏(w))∩[X,Y[.

  
  Initial step.
  Suppose that G(p)=∅. It follows
  from (<ref>)
  that Γ(q)=Γ(p) for every q∈(𝔏(w))∩[X,Y[.
  Therefore, [X,Y[ is a special bridge, by
  Lemma <ref>.
  
  Inductive step.
  Suppose that for every
  stationary point q such that
  |G(q)|<|G(p)|, there is a special bridge containing q.
  Consider the sets
  
    M   ={q∈(𝔏(w))∩[X,Y[:Γ(q)<Γ(p)},
    
        N   ={q∈(𝔏(w))∩[X,Y[:Γ(q)=Γ(p)}.

  Note that M∪ N=(𝔏(w))∩[X,Y[,
  by (<ref>).
  Note also that, by the induction hypothesis,
  every element of M is contained in some special bridge,
  as Γ(q)<Γ(p) implies G(q)⊊ G(p)
  with Γ(q)∈ G(p)∖ G(q).
  Our goal is to apply Lemma <ref>
  to the interval [X,Y[. So,
  it remains to show that every element of N is contained in some
  special bridge.
  For that purpose, we prove the following lemma.  
  
  
    The set N is closed.
  
  
  
    Consider a sequence (q_n)_n of elements of N
    converging to q. Since
    [X,Y[ and
    (𝔏(w)) are closed (the latter in view of
    Proposition <ref>), we only need to prove
    that Γ(q)=Γ(p).
    
    By
    Lemma <ref><ref>,
    we have F_g(q_n)⊆ F_g(q)
    for sufficiently large n.
    By the definition of N,
    we have F_g(q_n)= F_g(p) for all n,
    whence F_g(p)⊆ F_g(q).
    On the other hand, since
    q∈[X,Y[
    and thus q∈ I, by the choice of I
    (cf. Lemma <ref><ref>)
    we have F_g(q)⊆ F_g(p).
    This concludes the proof of the equality F_g(q)=F_g(p).
    
    By Lemma <ref>,
    we have Υ (q)≤_ JΥ (q_n)
    for some sufficiently large n.
    By the definition of N,
    we have Υ (p)=Υ (q_n)
    for all n, whence Υ (q)≤_ JΥ (p).
    On the other hand,
    since q∈ [X,Y[ we have Υ (p)≤_ JΥ (q)
    by (<ref>).
    Therefore we have Γ(p)=Γ(q).
  

  Let us proceed with the proof of Proposition <ref>.
    Suppose that q∈ N.
  Let I' be a clopen interval containing q,
  and contained in [X,Y[,
  satisfying the properties described in
  Lemmas <ref>
  and <ref><ref>.
  In a similar way as we have built X,Y from I,
  we may build from I' elements X',Y'∈ O_w such that
  q∈[X',Y'[ and
  (𝔏(w))∩[X',Y'[⊆ I'.
  
  Let x and y be two step points of [X',Y'[ such that
  x<y. We want to prove that the edge x y
  is g-projected.
  By Remark <ref>,
  we may as well assume that x and y are not ≈-equivalent.
  
  Suppose N∩ [x,y]=∅. Then every element of
  the nonempty intersection (𝔏(w))∩[x,y[
  belongs to M and is therefore
  contained in a special bridge, as observed earlier.
  Hence [ O_w(x), O_w(y)[ is a special bridge
  by Lemma <ref>.
  It follows that the edge x y is
  g-projected (cf. Remarks <ref>
  and <ref>).
  
  Suppose that N∩ [x,y]≠∅.
  Then
  N∩ [x,y] has a minimum r_x and a maximum r_y,
  because N∩ [x,y] is closed by Lemma <ref>.

 
  Let t be a J-minimum transition from
  r_x to r_y.
  Then we have t≤_ J J_r_x by Lemma <ref>,
  thus φ_ A(t)≤_ JΥ(r_x).
  On the other hand, r_x,r_y∈ I', and so, by the choice of I'
  (cf. Lemma <ref>),
  we know that φ_ A(t)≥_ JΥ(q).
  Since by the definition of N, the equalities
  Υ(r_x)=Υ(r_y)=Υ(q) hold,
  we obtain φ_ A(t)∈Υ(q).
  Also by the definition of N,
  we conclude that F_g(r_x)=F_g(r_y).
  This shows that (r_x,r_y) is a J-bridge.

  We claim  that [x,r_x[ is a bridge.
  Let x_1 and x_2 be step points such that x<x_1≤ x_2<r_x.
  We want to show that x_1 x_2 is g-projected,
  for which we may as well assume that x_1 and
  x_2 are not ≈-equivalent.
  Recall that M∪ N=(𝔏(w))∩[X,Y[
  by (<ref>), and so,
  by the definition of r_x,
  the set
  (𝔏(w))∩[ O_w(x_1), O_w(x_2)[
  is contained in M.
  But we already observed that every element of M is contained in some
  special bridge.
  Hence, [ O_w(x_1), O_w(x_2)[ is a special bridge
  by Lemma <ref>.
  and so x_1 x_2 is g-projected ((cf. Remarks <ref>
  and <ref>)), thus
  proving the claim. Similarly, ]r_y,y] is a bridge.

  By Proposition <ref>,
  there are step points x_0 and y_0
  satisfying x<x_0<r_x and r_y<y_0<y
  such that  x_0 y_0 is g-projected.
  Again
  by the definition of r_x and by (<ref>),
  every element of
  (𝔏(w))∩[ O_w(x), O_w(x_0)[
  belongs to M, and so [ O_w(x), O_w(x_0)[
  is a special bridge
  by Lemma <ref>.
  Hence, the edge 
  x x_0 is g-projected.
  Similarly, y_0 y is g-projected.
  It follows that x y is g-projected by
  Remark <ref>.
  
  We showed in all cases that
  x y is g-projected. Hence,
  [X',Y'[ is a special bridge containing q.

  We have established that every element of (𝔏(w))∩[X,Y[
  is contained in some special bridge.
  From Lemma <ref>,
  we then deduce that [X,Y[ is a special bridge containing p.
  This concludes the inductive step in our proof, thereby showing
  that the proposition holds.



  Let w be an element of AV.
  Let φ A→ S be a generating mapping of a finite aperiodic
  unambiguous semigroup S.
  Suppose that 𝔏_c(w)
  is g-recognized by (φ,s).
  Then 𝔏(w) is a bridge.



  Clearly, if w∈ A^+ then
  𝔏(w) is a bridge by
  Remark <ref>.
  Suppose that w∈ AA∖ A^+.
  By Proposition <ref>,
  for each stationary point p of 𝔏(w),
  there is a special bridge [X_p,Y_p[ containing it.
  The union of the nonempty family ([X_p,Y_p[)_p∈(𝔏(w))
  is 𝔏(w)∖{(w,1)}, therefore 𝔏(w)
  is a bridge, in view of Lemma <ref>.




  The direct implication in Theorem
  <ref> follows directly
  from Proposition <ref>.
  
  Conversely, suppose that
  𝔏(w) is g-recognized by (φ,s),
  for some g. Then (1,w) (w,1) is g-projected
  to g(1,w) g(w,1)
  by Theorem <ref>.
  But g(1,w)=(1,s)
  and g(w,1)=(s,1). It follows that s=φ_ A(w).


  


§ THE EFFECT OF MULTIPLICATION ON THE QUASI-ORDER




In this section, under suitable conditions, we relate
𝔉(uv) on one hand, with
𝔉(u) and 𝔉(v) on the other hand.

The quasi-orders considered in this section are all total, as they
stem from the quasi-order of 2-factorizations of equidivisible
semigroups. We want to compare different intervals of
quasi-ordered sets of 2-factorizations. This leads us to introduce
the following definitions.
Let (P,≤) and (Q,≤)
be two quasi-ordered sets,
and let φ be a function from P to Q.
Recall that φ is monotone if
p≤ q implies φ(p)≤φ(q), for every p,q∈ P.
Suppose moreover that the quasi-order on P is total.
Then we say that φ is
a quasi-isomorphism if φ
is a surjective monotone mapping such that,
for all p,q∈ P, we have
p< q⇒φ(p)<φ(q).
Because the quasi-order on P is total,
we have φ(p)<φ(q)⇒ p<q
and
φ(p)∼φ(q)⇒ p∼ q, for all p,q∈ P.
Therefore, φ induces the isomorphism
of linearly ordered sets
φ̃ P/∼→ Q/∼ sending p/∼ to φ(p)/∼.
In particular, the quasi-order on Q is also total.

Let I_u be an interval of 𝔏(u)
and I_v an interval of 𝔏(v), for some u,v
in a compact semigroup S.
A mapping θ I_u→ I_v
is said to be J-preserving
if J_p=J_θ(p)
for every p∈ I_u.


  
  Consider an equidivisible profinite semigroup S which is finitely
  cancelable, and take w∈ S.
  Let (u,v)∈𝔉(w). For p=(u,v)/∼,
  let e be the unique
  idempotent of J_p stabilizing (u,v). Then the following mappings
  are quasi-isomorphisms of intervals of the respective
  totally quasi-ordered sets 𝔉(w), 𝔉(u), and 𝔉(v):
  
    λ_(u,v)[(1,u),(u,e)]   →[(1,w),(u,v)],
    
        (x,y)   ↦(x,yv)
    ρ_(u,v)[(e,v),(v,1)]   →[(u,v),(w,1)].
    
        (x,y)   ↦(ux,y)
  
  Moreover, the induced isomorphisms
  λ̃_(u,v)
  and
  ρ̃_(u,v)
  are J-preserving.


Before proceeding with the proof of Proposition <ref>,
let us recall that the uniqueness of e mentioned in its statement
is guaranteed by Lemma <ref>.
The next technical lemma will be used
in the proof of Proposition <ref>.


  
  Let S be an equidivisible compact semigroup,
  and let x,y,z∈ S.
  If xy=xyz,
  then there exists some idempotent e∈ S^I such that
  yz^ω=ey and xe=x.
  Dually, if yz=xyz, then there exists some idempotent
  e∈ S^I such that x^ω y=ye and ez=z.



  We deal only with the case xy=xyz, as the other case is dual.
  Since S is equidivisible, the
  pairs (x,y) and (x,yz) are comparable
  elements of the quasi-ordered set 𝔉(xy).
  If (x,yz)≤(x,y), then there exists
  t∈ S^I such that x=xt and yz=ty, and so yz^k=t^ky
  for every k≥1, whence yz^ω=t^ω y and we choose
  e=t^ω. Otherwise,
  (x,y)<(x,yz), and so there exists u∈ S such that x=xu and
  y=uyz, whence x=xu^ω and y=u^ω yz^ω;
  since yz^ω=y=u^ω y, we may choose e=u^ω.



 
  By symmetry, it suffices to consider the mapping
  λ=λ_(u,v). 
  By Remark <ref>,
  and since ev=v, the mapping λ
  indeed takes its values in the interval [(1,w),(u,v)]
  and it is monotone.

  Let (x,z)∈𝔉(w) be such that (x,z)≤(u,v).
  Then there is t∈ S^I such that
  xt=u and z=tv.
  And, since u=ue,
  we deduce that (x,te)(u,e) is and edge
  of 𝒯(u), whence (x,te) belongs to
  the interval [(1,u),(u,e)].
  As λ(x,te)=(x,tev)=(x,z), we conclude
  that λ is surjective.

  To prove that λ is a quasi-isomorphism,
  it remains to show that
  if (x_1,y_1), (x_2,y_2) are elements of [(1,u),(u,e)], then
  
    (x_1,y_1)<(x_2,y_2)  (x_1,y_1v)<(x_2,y_2v).

  
  Reasoning by reductio ad absurdum,
  suppose that the implication fails, that is,
  that (x_1,y_1)<(x_2,y_2) and
  (x_2,y_2v)≤ (x_1,y_1v). 
  We may then consider s,t∈ S^I such that
  x_1t=x_2, y_1=ty_2, x_2s=x_1, and y_2v=sy_1v.
  The latter equality can be written as y_2v=st· y_2v,
  and applying the second case of Lemma <ref>
  to it, we conclude that there exists an idempotent
  f∈ S^I such that (st)^ω y_2=y_2f and fv=v. The
  calculations
  
    x_2·(st)^ω-1s=x_1(ts)^ω=x_1
       and  (st)^ω-1s· y_1=(st)^ω y_2=y_2f

  show that x_2· y_2f=x_1y_1=u and
  
    (x_2,y_2f)≤(x_1,y_1)

  in [(1,u),(u,e)].
  Since (x_1,y_1)<(x_2,y_2), we reach a contradiction provided
  we prove that y_2f=y_2.

  Recall that v=fv, and note that u=x_2y_2f implies u=uf.
  Hence, as (u,v)(u,v) belongs to
  the ideal 𝒦_p, we conclude
  that (u,v)(u,v) also belongs to
  𝒦_p.
  From Corollary <ref>,
  we get efe=e.
  On the other hand, since (x_2,y_2f)≤ (u,e)
  (cf. (<ref>)) and
  (x_2,y_2)≤(u,e), we have y_2fe=y_2f and
  y_2e=y_2. Hence, y_2f=y_2 fe=y_2efe=y_2e=y_2, and so we reach
  the desired contradiction.
  The contradiction was originated by the assumption that
  the implication (<ref>) fails in the interval
  [(1,u),(u,e)]. Hence, the implication holds, which concludes the
  proof that λ is a quasi-isomorphism.

    It remains to show that, for
    (x,y)∈ [(1,u),(u,e)], we have J_q=J_λ̃(q), where
    q=(x,y)/∼. Let ε∈ S^I be an idempotent.
    Observe that if ε stabilizes (x,y)
  then it also stabilizes (x,y v),
  which shows that J_q lies J-above J_λ̃(q).

  Conversely, suppose that ε stabilizes (x,yv).
  As ε yv=yv,
  it follows from Lemma <ref> that
  there is some idempotent
  f∈ S^I with  ε y=yf and fv=v.
  
  We claim that (x,yf)≤ (u,e). Suppose
  on the contrary that
  
    (u,e)<(x,yf).

  Then e≤_ Lyf, and so in particular we have e=ef.
  Similarly, from
  
    (x,y)≤ (u,e)

 we get y=ye. Therefore, yf=yef=ye=y,
 yielding a contradiction
 between (<ref>)
 and (<ref>).
 This shows the claim that (x,yf)≤ (u,e).

 We are now assured that (x,yf) belongs to the domain of
 λ. From v=fv, we get λ(x,y)=(x,yv)=λ(x,yf).
  We have
  already proved that λ is a quasi-isomorphism, and so we
  conclude that (x,y)∼(x,yf).
  On the other hand, ε clearly
  stabilizes (x,yf)=(x,ε y). Hence, J_λ̃(q)
  lies J-above J_q, which proves that the two J-classes
  coincide.



  Notice that in Proposition <ref>, in the special case
where (u,v) is a step point, we have e=1 and so the domains of
λ_(u,v) and ρ_(u,v) are, respectively,
𝔉(u) and 𝔉(v).



  Suppose that in
  Proposition <ref>
  we have e≠ 1. Let q be the stationary point (e,e)/∼ of 𝔏(e).
Since e stabilizes (e,e) and J_q is J-above e,
we have e∈ J_q. Therefore,
applying Proposition <ref>, we may consider the
quasi-isomorphisms
λ_(e,v)[(1,e),(e,e)]→[(1,v),(e,v)]
and
ρ_(u,e)[(e,e),(e,1)]→[(u,e),(u,1)].


The diagram in Figure <ref> may
facilitate the understanding of the applications of
Proposition <ref> in the case in which (u,v) is a stationary point.

The arrows indicate quasi-isomorphisms between various intervals of
the quasi-ordered sets 𝔉(w), 𝔉(u),
𝔉(v), and 𝔉(e). Those quasi-isomorphisms induce
isomorphisms between the corresponding intervals of the linearly
ordered sets 𝔏(w), 𝔏(u), 𝔏(v),
and 𝔏(e). The picture is perhaps clearer if interpreted
in this context, in which case, the points (u,v), (u,e), (e,v),
and (e,e) should be replaced by their respective ∼-classes.

 Let S be an equidivisible profinite semigroup S which is finitely
 cancelable, and let w∈ S.
 We endow every ordered subset Q of 𝔏(w)
 with the following labeling :
   for a step point p=(u,v) of 𝔏(w) belonging to Q,
   let (p)=(v) (and so
   if Q=𝔏(w) then the labeling on step points is the one defining the cluster word
   𝔏_c(w));
   for a stationary point p of 𝔏(w) belonging to Q,
   let (p)=J_p. The resulting labeled ordered set is denoted Q_.
   In the next result, P_+Q_ denotes
   the labeled ordered set with underlying
   ordered set P+Q and labeling whose restriction to P
   and Q is respectively the labeling of P_ and Q_.
   The symbol ≅ stands for isomorphism of labeled ordered sets.  


  
  Let S be an equidivisible profinite semigroup S which is finitely
cancelable. Take w∈ S. Let u,v∈ S be such that w=uv.
If e is the unique idempotent of J_(u,v)/∼ stabilizing (u,v), then

    𝔏(w)_≅[(1,u)/∼, (u,e)/∼[_+
      [(e,v)/∼,(v,1)/∼]_.

 In particular, if (u,v) is a step point, then
 𝔏(w)_≅(𝔏(u)∖{(u,1)})_+𝔏(v)_.



  Consider the quasi-isomorphism λ_(u,v) and respective
  isomorphism λ̃_(u,v) as in
  Proposition <ref>. Then, the pair (x,y)∈𝔉(u) is a step point of the interval
  [(1,u),(u,e)[ if and only if its image
  λ_(u,v)(x,y)=(x,yv) is a step point of
  [(1,w),(u,v)[, and (y)=(yv). This fact, together
  with λ̃_(u,v) being J-preserving, enables us to
  conclude that
  
    [(1,u)/∼,(u,e)/∼[_≅[(1,w)/∼,(u,v)/∼[_.

  Similarly, we obtain
  
    [(e,v)/∼,(v,1)/∼]_≅[(u,v)/∼,(w,1)/∼]_.

   Since we clearly have
  
    𝔏(w)_=[(1,w)/∼,(u,v)/∼)[_+[(u,v)/∼,(w,1)/∼]_,

  this concludes the proof.  





§ THE IMAGE OF THE REPRESENTATION
  W|->LC(W) IN THE
  APERIODIC CASE



Consider a cluster word (L,ℓ) over A.
Let φ A→ S be a generating mapping
of a semigroup S.
Let s∈ S and g(L)→𝔉(s) be such that
(L,ℓ) is g-recognized by (φ,s). We say that
g is a (φ,s)-recognizer of (L,ℓ).


  Let φ A→ S be a generating mapping of a finite semigroup S,
  and let π S→ T be an onto homomorphism of semigroups.
  Suppose that (L,ℓ) is recognized by (φ,s).
  Then (L,ℓ) is recognized by (π∘φ,π(s)).



  Let g(L)→𝔉(s)
  be a (φ,s)-recognizer of (L,ℓ).
  For each p∈(L), let g(p)=(u_p,v_p).
  Consider the mapping h(L)→𝔉(π(s))
  defined by h(p)=(π(u_p),π(v_p)). We claim
  that (L,ℓ) is h-recognized by (π∘φ,π(s)).
  The conditions <ref>-<ref>
  in Definition <ref>
  for h-recognition by  (π∘φ,π(s)) are
  clearly satisfied.
  It remains to show that condition <ref> holds.
  Let p be a stationary point of (L,ℓ).
  Take an element q of 𝔉(π(s)) such that
  h^-1(q) is left cofinal at p.
  Consider the set
  
    X={(u,v)∈𝔉(s):(π(u),π(v))=q}.

  Then we have h^-1(q)=g^-1(X)=⋃_x∈ X g^-1(x).
  Since h^-1(q) is left cofinal at p and
  X is finite, there is at least one element x_0 of
  X such that g^-1(x_0) is left cofinal
  at p. But then g^-1(x_0) is also right cofinal at p, because
  (L,ℓ) is g-recognized by (φ,s).
  Therefore, h^-1(q) is right cofinal at p.
  Symmetrically, if
  h^-1(q) is right cofinal at p,
  then it is left cofinal at p.
  This concludes our proof.  


For a cluster word (L,ℓ) over A, if p and q are step points of L such that p≤ q, then ([p,q],ℓ) is the
cluster word obtained from (L,ℓ) by restricting ℓ to [p,q[, and letting
ℓ(q)=1.

We wish to study cluster words (L,ℓ) satisfying the following conditions:



  * For every
  finite aperiodic unambiguous A-generated semigroup S,
  and every generating mapping φ A→ S, there is
  a unique s∈ S such that (L,ℓ)
  is recognized by (φ,s).

  * If p and q are step points of L
  such that p<q,
  then ([p,q],ℓ) satisfies <ref>.

  * Consider an arbitrary finite aperiodic unambiguous
  A-generated
  semigroup S
  and a generating mapping φ A→ S.
  Let s be such that (L,ℓ)
  if recognized by (φ,s).
  Take a (φ,s)-recognizer g.
  Suppose p and q are step points such that p<q.
  If t∈ S is such that (φ,t) recognizes ([p,q],ℓ),
  then g(p)g(q) is an edge of T(s).  
  




  In the setting of condition <ref>, there is only one such (φ,s)-recognizer, assuming that <ref> and <ref> also hold.
Indeed, if g is a (φ,s)-recognizer,
and p is a step point of L, and if s_1 and s_2
   are (the unique) elements
   of S such that ([min L,p],ℓ) and
   ([p,max L],ℓ) are respectively
   recognized by (φ,s_1) and (φ,s_2),
   then g(min L) g(p)
   and
   g(p) g(max L)
   are edges of T(s), and so g(p)=(s_1,s_2).




 Finally we consider a fourth condition, assuming 
  <ref>-<ref> hold:
   

  

  * For every step point p of L,
  there is a finite aperiodic unambiguous semigroup S
  and a generating mapping φ A→ S
  such that, for the unique (φ,s)-recognizer g of (L,ℓ),
  there are no elements of S
  that stabilize g(p) in T(S).  


A cluster word satisfying conditions
<ref>-<ref>
is called a worthy cluster word.


  A cluster word (L,ℓ) over A is
  isomorphic to a cluster word of
  the form 𝔏_c(w), w∈ AA,
  if and only if it is a worthy cluster word.



  Let w∈ AA. By Theorem <ref>,
  the cluster word 𝔏_c(w) satisfies
  condition <ref>.

  Take two step points p and q of 𝔏_c(w) such that p<q.
  Let t∈ AA be the unique transition from p to q.
  Applying twice Proposition <ref>, we conclude that
  ([p,q],ℓ) is isomorphic with 𝔏_c(t). 
  Therefore, by Theorem <ref>,
  ([p,q],ℓ) satisfies condition <ref>.

  By the previous paragraph and by Theorem <ref>,  the cluster word 𝔏_c(w) satisfies
  condition <ref>.
  
  Suppose condition <ref> does not hold for
  𝔏_c(w). Then, there is a step point p such that, for
  every finite aperiodic unambiguous A-generated semigroup S and
  every generating mapping φ A→ S, the vertex g(p)
  of 𝒯(s) is stabilized by some element of S. Let
  p=(u,v). We then have g(p)=(φ_ A(u),φ_
    A(v)) (cf. Theorem <ref>). By a
  standard compactness argument, this implies that (u,v) is
  stabilized by some element of AA. In view of
  Proposition <ref>, this is impossible
  since p is a step point.

  Conversely, suppose that (L,ℓ) is a worthy cluster word over A.
  Let (π_i)_i∈ I be an inverse system of continuous 
  homomorphisms π_i AA→ S_i onto finite aperiodic
  unambiguous A-generated semigroups, with connecting homomorphisms
  π_j,i S_j→ S_i, such that AA=_i∈ I S_i.
  According to condition <ref>,
  for each i∈ I, we may consider the unique element s_i of S_i such that
  (L,ℓ) is recognized by (π_i,s_i).
  Applying Lemma <ref>,
  we then conclude that π_j,i(s_j)=s_i, whenever i,j∈ I are such that
  i≤ j.
  Hence, we may consider the unique element w_L of AA
  such that
  
    π_i(w_L)=s_i,

  for every i∈ I.
  
    Consider the mapping
  λ(L)→(𝔏(w_L)) defined by
  
    λ(p)=(w_[min L,p],w_[p,max L]).

  Note that condition <ref>
  ensures that w_[min L,p] and w_[p,max L] are well defined.
  We claim that λ is an isomorphism
  between the cluster words (L,ℓ) and 𝔏_c(w_L).
  In the process of proving this
  we show that (w_[min L,p],w_[p,max L])
  is indeed a step point of 𝔉(w_L)
  (and thus of 𝔏(w_L)).
  
  We begin by observing
  that formula (<ref>) generalizes to
  every generating mapping φ A→ S of
  a finite aperiodic unambiguous A-generated semigroup.
  Indeed, take s∈ S such that (L,ℓ) is recognized by (φ,s).
  There is some i∈ I for which
  there is an onto homomorphism
  ρ S_i→ S satisfying φ_ A=ρ∘π_i.
  By Lemma <ref>, we know that
  ρ(s_i)=s. Therefore, we have
  
    φ_ A(w_L)=s.

  For such a pair (φ,s), let g_φ(L)→𝔉(s)
  be the unique (φ,s)-recognizer of (L,ℓ).
  If p is a step point of L, then applying formula
  (<ref>)
  to [min L,p] and to
  [p,max L], and taking into account
  Remark <ref>, we conclude that
  
    g_φ(p)=(φ_ A(w_[min L,p]),φ_ A(w_[p,max L])).

In particular,
we have

    φ_ A(w_[min L,p]w_[p,max L])=s=φ_ A(w_L).

Because φ was arbitrarily chosen among generating mappings of finite
aperiodic unambiguous A-generated semigroups, this shows
that the pair λ(p)=(w_[min L,p],w_[p,max L]) indeed belongs to 𝔉(w_L).
  
  Consider step points q and r
  of L such that q≺ r. Let a=ℓ(q).
  Take a generating mapping φ A→ S
  of a finite aperiodic unambiguous semigroup S.
  By the definition of (φ,s)-recognizer,
  we can consider in T(s) the edge
  g_φ(q)g_φ(r).
  In view of formula (<ref>), applied to q and r,
  we then have
  
    φ_ A(w_[min L,q]a)=φ_ A(w_[min L,r])  and  φ_ A(w_[q,max L])=φ_ A(aw_[r,max L]).

  Since φ was arbitrarily chosen among generating mappings of finite
  aperiodic unambiguous A-generated semigroups, we conclude that
  
    w_[min L,q]a=w_[min L,r]  and  
      w_[q,max L]=aw_[r,max L],

  that is, λ(q)λ(r)
  is an edge of T(w_L).
  By Proposition <ref>,
  we either have λ(q)∼λ(r) or λ(q)≺λ(r).
  
  If λ(q)∼λ(r), then
  there is z∈ a( AA)^I
  such that
  λ(q)λ(q)
  is an edge of T(w_L).
  Therefore, in view of formula (<ref>),
  we conclude that φ(z) labels a loop of T(s)
  rooted at g_φ(q).
  This contradicts the assumption
  that condition <ref> holds.

  We then conclude that, for step points q,r of L, we have
  
    q≺ rλ(q)≺λ(r).

  We also showed that ℓ(λ(q))=ℓ(q),
  thus establishing that
  the mapping λ(L)→(𝔏(w_L))
  has a well-defined codomain and that it preserves labels.

 Notice that formula (<ref>)
 can now be seen as follows: for
 the (φ,s)-recognizer
 g_w_L,φ(𝔏(w))→𝔉(s)
 of 𝔏_c(w_L),
 as in Proposition <ref>,
 we have
 
    g_φ(p)=g_w_L,φ(λ(p)),

 for every step point p of L.
  
  Let q and r be step points such that q<r.  
  Suppose that λ(q)≥λ(r).
  Then, for the pair (φ,s) considered so far,
  and in view of (<ref>),
  we may consider in T(s)
  an edge g_φ(r)g_φ(q) labeled by some
  t∈ S^I.
On the other hand, according to condition <ref>,
there is in T(s) an edge
g_φ(q)g_φ(r) labeled by some z∈ S.
   It follows that there is a loop in T(s)
   at g_φ(q)
   labeled by zt∈ S.
   This contradicts <ref>. Hence, we have
   λ(q)<λ(r).
   
   It remains to show that λ is onto.
   Let (u,v) be a step point of 𝔏(w_L).
   Consider the set 
   
    X={q∈(L):λ(q)≤ (u,v)}.

   Notice that X is nonempty: indeed, one clearly has min L∈ X.

   We claim that p=sup X is a step point.
   Suppose not.
   Let φ A→ S
   be the generating mapping of a finite aperiodic unambiguous A-generated
   semigroup. 
   Since g_φ is finite
   and {q∈(L):q>p} is right cofinal at p, there is
   (s_1,s_2)∈ g_φ
   such that
   
    R={q∈(L):q>p and g_φ(q)=(s_1,s_2)}

   is right cofinal at p.
   In particular, g_φ^-1(s_1,s_2) is right cofinal at p.
   Taking into account condition <ref>
   in Definition <ref>,
   we know that g_φ^-1(s_1,s_2) is also left cofinal at p.
   Therefore, there is a step point q such that q<p and
   g_φ(q)=(s_1,s_2). Since p=sup X, there is a step point
   q' such that q<q'<p and λ(q')≤ (u,v).
   We have already shown that λ is injective and respects the order,
   so we actually have λ(q)<(u,v).
   Let r be an element of the nonempty set R. Since r>p,
   we have (u,v)<λ (r). Let t_1 and t_2 be (the unique)
   transitions from λ(q) to (u,v) and from (u,v) to
   λ(r), respectively. Then, in T(s), we have the
   following edges
   
    g_w_L,φ(λ(q))
         g_w_L,φ(u,v)g_w_L,φ(λ(r)).

   But we have g_w_L,φ(λ(q))=g_φ(q)=(s_1,s_2)=g_φ(r)
    =g_w_L,φ(λ(r)).
   Hence, we can multiply the second edge in (<ref>)
   with the first edge, obtaining a loop at
   g_w_L,φ(u,v)=(φ_ A(u),φ_ A(v))
   labeled by φ_ A(t_2t_1)∈ S,
   leading to a contradiction, since 𝔏_c(w_L) satisfies <ref>. 
   This establishes the claim that p is a step point, thus p∈ X.

   Suppose that λ(p)<(u,v).
   Let p' be the step point such that p≺ p'. Then,
   applying (<ref>),
   we get λ(p)≺λ (p'). 
   Since λ(p)< (u,v), we
   obtain λ(p')≤ (u,v), and so p'∈ X.
   But then p'≤sup X=p, a contradiction with p<p'.
   As p∈ X, to avoid the contradiction, we must have λ(p)=(u,v).
   This concludes the proof that λ is onto. 


It would also be interesting to characterize the worthy clustered
linear orders that arise as the images 𝔏_c(w) of
ω-words w. We leave this as an open problem.



§ ON THE CARDINALITY OF THE SET OF STATIONARY POINTS



Let V be an equidivisible pseudovariety of semigroups not contained
in CS. Then V is finitely cancelable
(cf. Proposition <ref>),
and so, by Theorem <ref>, for a finite alphabet A and for w∈ AV,
the set (𝔏(w)) of step points is
the set of isolated points of
𝔏(w), with respect to the order topology.
Therefore, (𝔏(w)) is at most countable
by Corollary <ref>.
The aim of this section
is to show that when A has at least two elements,
there are elements w in AV
for which the set  (𝔏(w)) of stationary points
has cardinal 2^ℵ_0. This will be done
using some tools originated from
symbolic dynamics, following an approach that
has been successfully used in recent years
to elucidate structural aspects of
relatively free profinite semigroups <cit.>.


 

 §.§ Subshifts

 Consider a finite alphabet A,
 and endow A^ℤ
 with the product topology, where A is endowed with
 the discrete topology.
 The shift map of A^ℤ is the
 homeomorphism σ A^ℤ→ A^ℤ,
 defined by σ((x_i)_i∈ℤ)=(x_i+1)_i∈ℤ.
  A symbolic dynamical system of  A^ℤ,
  also called subshift of A^ℤ,
  is a nonempty closed subset X of A^ℤ
  such that σ( X)= X.
  The books <cit.>
  are good references on symbolic dynamical systems.

 
   We say that a subset L of a semigroup S is


  * factorial if it is closed under taking factors;

  * prolongable if, for every s∈ L, there are t,u∈
  S such that ts,su∈ L;

  * irreducible if, for all s,t∈ L, there is u∈ S
  such that sut∈ L.


If X is a subshift of A^ℤ, then L( X) denotes
the language of the words of A^+ of the form
x_kx_k+1… x_k+n, where
k∈ℤ, n≥ 0 and (x_i)_i∈ℤ∈ X.
The set L( X)
is a factorial and prolongable language of A^+,
and in fact all nonempty factorial and prolongable languages of A^+ are of
  this form; moreover, Y⊆ X
  if and only if L( Y)⊆ L( X),
  whenever X
  and Y are subshifts of
  A^ℤ <cit.>.
  Finally, X is said to be irreducible if  L( X)
 is an irreducible subset of A^+.

   If X is a subshift of A^ℤ then
   the sequence (1/nlog_2 |L( X)∩ A^n|)_n
   converges to its infimum, which is called the
   entropy of X and denoted h( X) <cit.>.
   Note that X⊆ Y     implies h( X)≤ h( Y),
   whenever X and Y are subshifts.


  If X is a subshift of A^ℤ
  then h( X)≤log_2|A|=h(A^ℤ).
  Moreover, from the fact that
  (1/nlog_2 |L( X)∩ A^n|)_n converges to its infimum
  one easily deduces that the subshift X of A^ℤ satisfies
  h( X)=log_2|A| if and only if X=A^ℤ
  (this a special case
  of <cit.>).




 §.§ A special J-class
    

Consider a subshift X of A^ℤ, and suppose that V
is a pseudovariety containing LSl. Let M_ V( X) be the
set of pseudowords w∈ AV such that all finite factors of w
belong to L( X). The set M_ V( X) is a factorial subset of AV. Because, as it is well known, the languages of the form A^∗
uA^∗, with u∈ A^+, are LSl-recognizable, the
hypothesis that V contains LSl ensures that M_ V(
X) is a closed subset of AV (cf. <cit.>).

    
      Let X be an irreducible subshift of
      A^ℤ.
      Consider a pseudovariety V containing LSl.
      For every u,v∈ M_ V( X) there is w∈ AV,
      depending only on the finite suffixes of u and
      on the finite prefixes of v,
      such that uwv∈ M_ V( X).      
    

    
      If V is a pseudovariety containing D and its dual, then
      every infinite element of AV has a unique prefix (suffix)
      in A^+ with length n, for every n≥ 1.
      Let s_n be the suffix
      of length n of u
      and let p_n be the prefix of length n of v.
      Since X is irreducible, 
      for each n∈ℕ, there is
      w_n∈ L( X) such that
      s_nw_np_n∈ L( X).
      Let w be an accumulation point
      of (w_n)_n.
      Then w has the desired property.
    

[<cit.>]
  
  Let S be a compact semigroup and let X⊆ S. Then X is a
  closed, factorial, irreducible subset of S if and only if X consists of
  all factors of some regular element of S.


By Lemma <ref>
and
Proposition <ref>,
if X is an irreducible subshift, then
there is a unique regular J-class J_ V( X)
such that the elements of M_ V( X) are the factors
of elements of J_ V( X).


It also follows from
Proposition <ref>
that J_ V( X)≤_ J J_ V( Y)
if and only if M_ V( Y)⊆ M_ V( X).
Since we clearly have M_ V( Y)⊆ M_ V( X)
if and only if L( Y)⊆ L( X),
we conclude that
    
    Y⊆ X
        J_ V( X)≤_ J J_ V( Y).
  


 
   If X=A^ℤ then M_ V( X)= AV, and so
   J_ V( X) is the minimum ideal of AV.
 

 

 §.§ Uncountable <R-chains
   and uncountable sets of stationary points


We use the standard notation ⌈α⌉ for the least
integer greater than or equal to the real number α.


  There is a family (𝒮_β)_β∈]1,+∞[ of
  symbolic dynamical systems, parameterized by the set of real numbers
  greater than one, such that:
  
  
  * S_β is an irreducible subshift of
    {0,…,⌈β⌉-1}^ℤ;
    
  
  * h(
    S_β)=log_2β;
  
  * for every α,β∈]1,+∞[, we have
    α<β if and only if S_α⊊
    S_β.
        


A concrete family of symbolic dynamical systems satisfying the
conditions of
Theorem <ref>
is the family of β-shifts. A comprehensive exposition
about this family can be found
in <cit.>
and <cit.>.
That these subshifts are irreducible follows from them being coded <cit.> —
a subshift X of A^ℤ is coded if there is a prefix
code Y contained in 
A^+ such that L( X) is the set of factors of elements of Y^+.
The entropy of β-shifts was computed
in <cit.>.
The fact that this class fits into
Property <ref>
of Theorem <ref>
appears at the beginning of <cit.>
(actually, only the implication
α≤β⇒ S_α⊆ S_β
is explicit there, but
from Property <ref>
one gets S_α⊊ S_β⇒α <β).

As usual, the notation <_ J stands for the irreflexive relation
originated by ≤_ J, and similarly for <_ R and <_ L.
From Theorem <ref>
   and equivalence (<ref>)
   in Remark <ref>
   one immediately deduces the existence of a <_ J-chain
   in AV formed by 2^ℵ_0 regular elements,
whenever V
contains LSl
and A has at least two letters.
The next theorem gives a refinement of this, as it shows in particular
the existence in AV of a <_ R-chain
formed by 2^ℵ_0 regular elements.
We remark that
in <cit.> an example is given of a <_ R-chain
of 2^ℵ_0 non-regular elements in ALSl, when |A|>1.


  Let V be a finitely cancelable pseudovariety of semigroups
  containing LSl and let (𝒮_β)_β∈]1,+∞[ be a family of subshifts as in
  Theorem <ref>. Fix an
  integer n>1 and let A be the alphabet {0,…,n-1}. There
  is a family (w^(β))_β∈]1,n] of pseudowords of
  AV satisfying the following conditions:
  
  
  * 
    w^(β)∈ J_ V( S_β)⊆AV,
    for every
    β∈]1,n];
  
  * 
    α<β⇔ w^(β)<_ Rw^(α),
        for every
    α,β∈]1,n];
  
  * 
    w^(n) is an element of the minimum ideal of AV;
  
  *  there is a subnet of
    (w^(β))_β∈]1,n[ converging to w^(n), where
    ]1,n[ is endowed with the usual order;
  
  *  for each
    β∈]1,n], there are v^(β),
    f^(β)
    such that (w^(β),v^(β)) is a stationary point of
    𝔉(w^(n)), 
    and, for q_β=(w^(β),v^(β))/∼,
    the pseudoword 
    f^(β) is an idempotent in
    J_q_β stabilizing (w^(β),v^(β))
    and satisfying f^(β) L w^(β);
  
  * 
    we have  α <β⇒ q_α<q_β,
    and if  moreover AV is equidivisible,
    then the equivalence α <β⇔
    q_α<q_β holds, for every α,β∈]1,n].
  



The proof of Theorem <ref>
will
be done in several steps.
But first
we highlight the following corollary, which is our main motivation for
the theorem.


  
  Let V be a finitely cancelable pseudovariety of semigroups
  containing LSl and
  let A be a finite alphabet with at least two elements.
  Then there are pseudowords w in the minimum ideal of AV
  such that (𝔏(w)) has 2^ℵ_0 elements.
  

Note also that Theorem <ref>
gives an example of a pseudoword in the minimum ideal of AV
whose set of stationary points contains a subset with the same order
type as the set of real numbers. In contrast, the following example
exhibits a pseudoword also in the minimum ideal of AV with only
one stationary point.


  Let u_1,u_2,u_3,… be an enumeration of the elements
  of A^+, and let V be a pseudovariety
  containing LSl such that AV is equidivisible.
  For each k≥ 1, consider in AV an accumulation
  point v_k
  of the sequence (u_ku_k+1⋯ u_n-1u_n)_n≥ k
  and an accumulation point w_k of the sequence
  (u_nu_n-1⋯ u_k+1u_k)_n≥ k.
  As every element of A^+ is a factor of v_k and w_k,
  we know that v_k and w_k belong to the minimum ideal K_A of AV.
  Therefore, if p and q are respectively the first
  and last stationary point of 𝔏(v_1w_1), then J_p=J_q=K_A by
  Theorem <ref>,
  and so p=q by Lemma <ref>.




 §.§ About the proof of Theorem <ref>

  

Let S be a compact semigroup and I an ordered set. Suppose that
F=(F_i)_i∈ I is a nonempty family of compact subsets of S.
Denote by R_F the set of partial functions f from I to ⋃
F such that f(i)∈ F_i for all i∈ f, and such that i≤
j ⇒ f(i)≤_ R f(j) whenever i,j∈ f. We
endow R_F with the partial order ≤ defined by

    f≤ g 
      (f=g f⊊ g).

    

  The ordered set R_F has a maximal element.



  Let C be a chain of elements of R_F. We want to show that C
  has an upper bound in R_F. For each f∈ R_F, let f' be an
  element of ∏_i∈ I F_i whose restriction to f equals
  f. Since ∏_i∈ IF_i is compact, the net (f')_f∈ C
  has a subnet converging to some φ∈∏_i∈ IF_i. For
  achieving our goal, we may as well assume that (f')_f∈ C
  converges. Let us fix an element f_0 of C, and take i,j∈
  f_0 such that i≤ j. For all f∈ C such that f_0≤ f,
  one has f(i)≤_ R f(j). As the net (f')_f∈ C
    f_0≤ f converges to φ and ≤_ R is a closed
  relation, we deduce that φ(i)≤_ Rφ(j).
  Moreover, since F_i is closed, we also have φ(k)∈ F_k
  for all k∈ f_0. As f_0 was chosen arbitrarily from C, we
  conclude that the restriction of φ to ⋃_f∈ C
  f belongs to R_F and is an upper bound for C. Hence, by
  Zorn's Lemma, R_F has a maximal element.

    

  
  For the relation ⊇, let C be a nonempty chain of
  irreducible subshifts of A. Consider a pseudovariety of
  semigroups V that contains LSl. Let 𝒥 be the family of
  J-classes (J_ V( X))_ X∈ C. Then there is an
  element of R_𝒥 with domain C.

  



  By Lemma <ref>, we know there is
  in R_𝒥 a maximal element f. We claim that f= C.
  Suppose this is false. Let Z∈ C∖ f.
  Supposing that I={ X∈ f:  X⊆ Z} is
  nonempty, let u be an accumulation point of the net (f(
  X))_ X∈ (I,⊆); in case I=∅, we let u be any
  element of J_ V( Z). Since X⊆ Z implies
  M_ V( X)⊆ M_ V( Z), we have f( X)∈ M_
  V( Z) for all X∈ I. And since M_ V( Z) is
  closed, we conclude that u∈ M_ V( Z). Moreover, fixed X∈ I, then, as f∈ R_𝒥, we have f( Y) ≤_ Rf(
  X) for all Y∈ I such that X⊆ Y, whence
  
    X∈ I u≤_ R f( X).

  Let v∈ J_ V( Z). By the irreducibility of M_ V(
  Z), there is w∈ AV such that uwv∈ M_ V( Z).
  Since uwv is a factor of v and v is a J-minimum element of
  M_ V( Z), we have uwv∈ J_ V( Z). As f∈ R_𝒥,
  every two elements in the image of f are R-comparable,
  and so the elements in the image of f have all the same set P of
  finite prefixes. By
  Lemma <ref>, for each
  X∈ f such that Z⊆ X, there is a
  pseudoword w', depending only on uwv and P, such that
  uwvw'f( X)∈ M_ V( X). More precisely, we have
  uwvw'f( X)∈ J_ V( X), as f( X)∈ J_ V( X).
  The partial function
  
    f' X∈ f
        ∪{ Z}↦
          f( X)    if  X⊊ Z,
    
          uwv    if  X= Z,
    
          uwvw'f( X)    if  Z⊊ X,

  belongs to R_𝒥 (cf.
  implication (<ref>)) and f⊊ f'. This contradicts the fact that f is a
  maximal element of R_𝒥. The absurdity comes from the hypothesis
  C∖ f≠∅.

  
We recall the concept of entropy of a pseudoword, first introduced
in <cit.>, and applied there in the study of
relatively free profinite semigroups. Some further applications were
given in <cit.>. Let V be a pseudovariety
containing LSl and A an alphabet with at least two letters.
For w∈ AV, let q_w(n) denote the number of factors of length
n of w. If w is an infinite pseudoword then the sequence
1/nlog_2 q_w(n) converges to its infimum, which is denoted
by h(w) and called the entropy of w.[This is the
  definition used in <cit.>.
  In <cit.> the entropy of w is defined as
  1/nlog_|A| q_w(n), which equals h(w)log_|A|2 for
  h(w) as defined here.] This definition extends to finite words,
by letting h(w)=0 when w is finite. If X is a subshift, then
h( X)=h(w) for every pseudoword w whose set of finite factors
is equal to L( X). For instance, if X is irreducible then
h( X)=h(w) when w∈ J_ V( X).
     

Note that h(w)∈[0,log_2|A|], for all w∈ AV. Moreover, we
have the following fact from <cit.>.
    

  Let w∈ AV. Then h(w)=log_2|A| if and only if w belongs
  to the minimum ideal of AV.

    
In particular, the entropy of pseudowords of AV is not
continuous, since every finite word has entropy zero and the set of
finite words is dense. However, it is upper semi-continuous, as proved
next.


  Let V be a pseudovariety containing LSl. If (w_n)_n
  is a sequence of elements of AV converging to w then
  lim sup h(w_n)≤ h(w).



  Since lim sup h(w_n) is the greatest accumulation point of the
  sequence (h(w_n))_n, the proof is reduced to the case where
  (h(w_n))_n converges.
    
  Since lim w_n=w and V contains LSl, for each k
  there is p_k such that for all n≥ p_k the pseudowords w_n
  and w have the same factors of length k. Let (n_k)_k be the
  sequence recursively defined by n_1=p_1 and
  n_k+1=max{n_k,p_k+1}. Given ε>0, consider the
  set K={k:h(w_n_k)≥ h(w)+ε}. For every k∈ K,
  one has
  
    1/klog_2 q_w(k)=1/klog_2 q_w_n_k(k)
        ≥ h(w_n_k)≥ h(w)+ε.

  As lim1/klog_2 q_w(k)=h(w), if K is infinite
  then (<ref>) leads to the
  contradiction h(w)≥ h(w)+ε. Hence K is finite, and
  so lim h(w_n_k)≤ h(w)+ε. Since ε is
  arbitrary and (h(w_n))_n converges, we conclude that lim
  h(w_n)≤ h(w).


We now have all the tools to achieve the proof of
Theorem <ref>.


  Let C be the chain (𝒮_β)_]1,n[, ordered
  by ⊇. Applying
  Proposition <ref> to the family of
  J-classes (J_ V( X))_ X∈ C, we conclude
  that there is a function f C→ AV such that f(
  S_β)∈ J_ V( S_β) and
  
    S_β⊇ S_α
        f( S_β) ≤_ Rf( S_α),
            ∀α,β∈]1,n[.

  On the other hand, if f( S_β) ≤_ Rf( S_α),
  then S_β⊇ S_α by
  equivalence (<ref>) in
  Remark <ref>. Therefore, we actually have
  
    S_β⊋ S_α 
        f( S_β) <_ Rf( S_α),
            ∀α,β∈]1,n[.

  For each β∈]1,n[, let w^(β)=f( S_β). By
  the given characterization of ( S_β)_β∈]1,+∞[
  (cf. Theorem <ref>), we
  know that S_β⊋ S_α if and only if
  α<β, whence (<ref>) translates to
  
    α<β w^(β)<_ Rw^(α),
            ∀α,β∈]1,n[.

  Let (α_k)_k be an increasing sequence of elements of the open
  interval ]1,n[ such that limα_k=n. Thus, we have lim
  h(w^(α_k))=limlog_2 α_k=log_2 n. Hence, if
  w^(n) is an accumulation point of the sequence
  (w^(α_k))_k, then h(w^(n))=log_2n by
  Lemma <ref> and
  Remark <ref>. By
  Proposition <ref>, the pseudoword
  w^(n) then belongs to the minimum ideal of AV. Since
  ≤_ R is a closed relation, we have w^(n)≤_ R
  w^(β) for all β∈]1,n[. Then, taking into
  account (<ref>), we conclude that the net
  (w^(β))_β∈]1,n] satisfies
  conditions <ref>-<ref>
  in Theorem <ref>.

  As w^(n)≤_ Rw^(β), there is u^(β) with
  w^(n)=w^(β)u^(β). Since w^(β) is regular,
  there is an idempotent f^(β) in the L-class of
  w^(β). Take v^(β)=f^(β)u^(β). Then
  (w^(β),v^(β)) is an element of 𝔉(w^(n)) stabilized by f^(β). By
  Proposition <ref>, this implies
  that q_β=(w^(β),v^(β))/∼ is a stationary
  point.
  The elements of J_q_β are factors of w^(β), and so
  by the minimality of J_q_β we have f^(β)∈
  J_q_β. Hence,
  condition <ref> in
  Theorem <ref> holds.

  Let α,β∈]1,n]. Since q_α≤
  q_β⇒ w^(β)≤_ R w^(α), we deduce
  from (<ref>) that q_α≤
  q_β⇒α≤β. Thus, if α<β
  then we cannot have q_β≤ q_α, and so assuming AV
  is equidivisible, we get q_α< q_β, thereby establishing
  condition <ref> in
  Theorem <ref>.





§ ACKNOWLEDGMENTS


The work of the first, third, and fourth authors was partly supported
by the Pessoa French-Portuguese project “Separation in
automata theory: algebraic, logical, and combinatorial aspects”.

The work of the first three authors was also partially supported
respectively by CMUP (UID/MAT/ 00144/2013), CMUC (UID/MAT/00324/2013),
and CMAT (UID/MAT/ 00013/2013), which are funded by FCT (Portugal) with
national (MCTES) and European structural funds (FEDER), under the
partnership agreement PT2020.


The work of the fourth author was partly supported by ANR 2010
BLAN 0202 01 FREC and by the DeLTA project ANR-16-CE40-0007.

amsplain






10

Almeida:1994a
J. Almeida, Finite semigroups and universal algebra, World Scientific,
  Singapore, 1995, English translation.

Almeida:2005c
, Profinite groups associated with weakly primitive
  substitutions, Fundamentalnaya i Prikladnaya Matematika (Fundamental and
  Applied Mathematics) 11 (2005), 13–48, In Russian. English version
  in J. Math. Sciences 144, No. 2 (2007) 3881–3903.

Almeida:2003cshort
, Profinite semigroups and applications, Structural theory of
  automata, semigroups and universal algebra (New York) (V. B. Kudryavtsev and
  I. G. Rosenberg, eds.), Springer, 2005, pp. 1–45.

Almeida   ACosta:2007a
J. Almeida and A. Costa, Infinite-vertex free profinite semigroupoids and
  symbolic dynamics, J. Pure Appl. Algebra 213 (2009), 605–631.

Almeida   ACosta:2013
, Presentations of Schützenberger groups of minimal
  subshifts, Israel J. Math. 196 (2013), no. 1, 1–31.

Almeida   ACosta:2016a
, Equidivisible pseudovarieties of semigroups, Publ. Math.
  Debrecen (2016), To appear, arXiv:1603.00330.

Almeida   Klima:2015a
J. Almeida and O. Klíma, Representations of relatively free
  profinite semigroups, irreducibility, and order primitivity, Tech. report,
  Univ. Masaryk and Porto, 2015, arXiv:1509.01389.

Almeida   Volkov:2001a
J. Almeida and M. V. Volkov, Profinite methods in finite semigroup
  theory, Proceedings of International Conference “Logic and applications”
  honoring Yu. L. Ershov on his 60-th birthday anniversary and of International
  Conference on mathematical logic, honoring A. I. Mal'tsev on his 90-th
  birthday anniversary and 275-th anniversary of the Russian Academy of
  Sciences (Novosibirsk, Russia) (S. S. Goncharov, ed.), 2002, pp. 3–28.

Almeida   Volkov:2006
, Subword complexity of profinite words and subgroups of free
  profinite semigroups, Int. J. Algebra Comput. 16 (2006), 221–258.

Almeida   Weil:1996b
J. Almeida and P. Weil, Free profinite R-trivial monoids, Int.
  J. Algebra Comput. 7 (1997), 625–671.

Bertrand-Mathis:1986
A. Bertrand-Mathis, Développement en base θ; répartition
  modulo un de la suite (xθ^n)_n≥ 0; langages codés et
  θ-shift, Bull. Soc. Math. France 114 (1986), no. 3,
  271–323.

Birget:1984
J.-C. Birget, Iteration of expansions—unambiguous semigroups, J. Pure
  Appl. Algebra 34 (1984), no. 1, 1–55.

Bruyere   Carton:2007
V. Bruyère and O. Carton, Automata on linear orderings, J. Comput.
  System Sci. 73 (2007), no. 1, 1–24.

Carruth   Hildebrant   Koch:1983
J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The theory of
  topological semigroups, Pure and Applied Mathematics, no. 75, Marcel Dekker,
  New York, 1983.

Chaubard   Pin   Straubing:2006
L. Chaubard, J.-E. Pin, and H. Straubing, Actions, wreath products of
  𝒞-varieties and concatenation product, Theor. Comp.
  Sci. 356 (2006), 73–89.

ACosta:2007t
A. Costa, Semigrupos profinitos e dinmica simblica, Ph.D. thesis,
  Univ. Porto, 2007.

ACosta   Steinberg:2011
A. Costa and B. Steinberg, Profinite groups associated to sofic shifts
  are free., Proc. London Math. Soc. 102 (2011), 341–369.

Costa:2001a
J. C. Costa, Free profinite locally idempotent and locally commutative
  semigroups, J. Pure Appl. Algebra 163 (2001), 19–47.

Eilenberg:1976
S. Eilenberg, Automata, languages and machines, vol. B, Academic Press,
  New York, 1976.

Frougny:2000
C. Frougny, Number representation and finite automata, Topics in
  symbolic dynamics and applications (Temuco, 1997), London Math. Soc.
  Lecture Note Ser., vol. 279, Cambridge Univ. Press, Cambridge, 2000,
  pp. 207–228.

Gool   Steinberg:2016
S. J. van Gool and B. Steinberg, Pro-aperiodic monoids via saturated
  models, Tech. report, 2016, arXiv:1609.07736.

Grillet:1995
P.-A. Grillet, Semigroups, An introduction to the structure theory,
  Monographs and Textbooks in Pure and Applied Mathematics, vol. 193, Marcel
  Dekker Inc., New York, 1995.

Huschenbett   Kufleitner:2014
M. Huschenbett and M. Kufleitner, Ehrenfeucht-Fraïssé games on
  omega-terms, LIPIcs. Leibniz Int. Proc. Inform., vol. 25, STACS, Schloss
  Dagstuhl. Leibniz-Zent. Inform., Wadern, 2014, pp. 374–385.

Ito   Takahashi:1974
S. Ito and Y. Takahashi, Markov subshifts and realization of
  β-expansions, J. Math. Soc. Japan 26 (1974), 33–55.

Kitchens:1998
B. Kitchens, One-sided, two-sided and countable state Markov shifts,
  Springer, Berlin, 1998.

Kufleitner   Wachter:2016
M. Kufleitner and J. P. Wächter, The word problem for omega-terms over
  the Trotter-Weil hierarchy (extended abstract), Computer
  science—theory and applications, Lecture Notes in Comput. Sci., vol. 9691,
  Springer, [Cham], 2016, pp. 237–250.

Lallement:1979
G. Lallement, Semigroups and combinatorial applications,
  Wiley-Interscience, J. Wiley & Sons, Inc., New York, 1979.

Lind   Marcus:1996
D. Lind and B. Marcus, An introduction to symbolic dynamics and coding,
  Cambridge University Press, Cambridge, 1995.

Lothaire:2001
M. Lothaire, Algebraic combinatorics on words, Cambridge University
  Press, Cambridge, UK, 2002.

McKnight   Storey:1969
J. D. McKnight, Jr. and A. J. Storey, Equidivisible semigroups, J.
  Algebra 12 (1969), 24–48.

Moura:2009a
A. Moura, Representations of the free profinite object over DA, Int.
  J. Algebra Comput. 21 (2011), 675–701.

Parry:1960
W. Parry, On the β-expansions of real numbers, Acta Math. Acad.
  Sci. Hungar. 11 (1960), 401–416.

Pin:2009S
J.-E. Pin, Profinite methods in automata theory, vol. 26th International
  Symposium on Theoretical Aspects of Computer Science (STACS 2009),
  Internationales Begegnungs- Und Forschungszentrum für Informatik (IBFI),
  2009, pp. 31–50.

Pin   Weil:2002b
J.-E. Pin and P. Weil, The wreath product principle for ordered
  semigroups, Comm. Algebra 30 (2002), no. 12, 5677–5713.

Renyi:1957
A. Rényi, Representations for real numbers and their ergodic
  properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477–493.

Rhodes   Steinberg:2002
J. Rhodes and B. Steinberg, Profinite semigroups, varieties, expansions
  and the structure of relatively free profinite semigroups, Int. J. Algebra
  Comput. 11 (2002), 627–672.

Rhodes   Steinberg:2009qt
, The q-theory of finite semigroups, Springer Monographs in
  Mathematics, Springer, 2009.

Rosenstein:1982
J. G. Rosenstein, Linear orderings, Academic Press, New York, 1982.

Schutzenberger:1965
M. P. Schtzenberger, On finite monoids having only trivial subgroups,
  Inform. and Control 8 (1965), 190–194.

Straubing:1979a
H. Straubing, Aperiodic homomorphisms and the concatenation product of
  recognizable sets, J. Pure Appl. Algebra 15 (1979), 319–327.

Thomas:1997
W. Thomas, Languages, automata, and logic, Handbook of formal languages.
  Beyond words (G. Rozenberg and A. Salomaa, eds.), vol. 3, Springer, Berlin,
  1997, pp. 389–455.

Tilson:1987
B. Tilson, Categories as algebra: an essential ingredient in the theory
  of monoids, J. Pure Appl. Algebra 48 (1987), 83–198.

Weil:2002
P. Weil, Profinite methods in semigroup theory, Int. J. Algebra Comput.
  12 (2002), 137–178.

Willard:1970
S. Willard, General topology, Addison-Wesley, Reading, Mass., 1970.



