Hypergroup Deformations of Semigroups
Vishvesh Kumar Kenneth A. Ross Ajit Iqbal Singh
math.FA math.CO 43A62, 20M14 (Primary)


 We view the well-known example of the dual of a countable compact hypergroup, motivated by the orbit space of p-adic integers by Dunkl and Ramirez (1975), as hypergroup deformation of the max semigroup structure on the linearly ordered set ℤ_+ of the non-negative integers along the diagonal. This works as motivation for us to study hypergroups or semi convolution spaces arising from “max" semigroups or general commutative semigroups via hypergroup deformation on idempotents. 



[2010]Primary  43A62, 20M14
[
    L. N. Pfeiffer
    December 30, 2023
=====================





§ INTRODUCTION

We introduce and study hypergroups, same as convolution spaces, in short, convos, or, semi convolution spaces, in short, semiconvos <cit.> arising from general commutative semigroups via deformation of the part of diagonal consisting of the idempotents. The genesis was the well-known example related to the orbit space of p-adic integers by Dunkl and Ramirez <cit.> viewed with this perspective on one hand and a good account of the structure of measure algebras of certain linearly ordered semigroups with order topology in <cit.> and <cit.> on the other hand (see also <cit.>). There is a substantial development of hypergroup deformations of groups and their applications. To get an idea one can see <cit.> and <cit.>. The present work can be seen as a complement of that because a group has no idempotents other than the identity. 


The next section gives basics of semigroups and hypergroups in the form that we need with a little touch of novelty at a few places. We begin Section 3 with an attempt to make a “max" semigroup (S, <, ·) with the discrete topology into a hermitian discrete hypergroup by deforming the product on the diagonal. Amongst other things, we arrive at the result that this can be done if and only if either S is finite or S is isomorphic to (ℤ_+,<,max). We determine its  dual and show that it becomes a countable compact hermitian  hypergroup with respect to pointwise multiplication. In Section 4, we prove our main theorem on semiconvo or hypergroup deformations of idempotents in commutative semigroups.

Let ℤ_+= ℕ∪{0} and ×, the usual multiplication. For a subset T of S, _T denotes the characteristic function of T defined on S. For notational convenience, we take empty sums to be zero. 


§ BASICS OF SEMIGROUPS AND HYPERGROUPS
 
For basics of semigroups and hypergroups one can refer to standard  books, monographs and research papers. For instance, one can see (<cit.>, <cit.>, <cit.>, <cit.> <cit.>, <cit.>, <cit.>) for semigroups and (<cit.>, <cit.>, <cit.>, <cit.>,  <cit.>, <cit.>, <cit.>, <cit.>) for hypergroups. However, we give below some of them in the form we need.


 §.§ Basics of semigroups
  
  (i)  As in <cit.>, we consider a non-empty set S linearly ordered by the relation `<'. 


  (a) For  m,n ∈ S, we define m · n= max{m,n}. This makes it into a commutative semigroup. In this paper, we call such a semigroup (S,<,·) a “max" semigroup. At times, we will write it as (S,<,max). Further, for m,n ∈ S, we write ℒ_n = { k ∈ S : k < n },  𝒰_m = {k ∈ S : m <k }.

  (b) The linear order `<' and the “max" semigroup operation as in (a) above are appropriately related in the sense that  m,n,k ∈ S and m≤ n together imply that mk≤ nk as specified in Definition 0.1 of linearly ordered semigroup in <cit.>. 

  (c) We may define another product `·' by min{m,n} but we stick to max unless needed for some explicit purpose.


  (ii) We call a commutative semigroup (S, ·) max-min type if for m,n ∈ S,  m· n is m or n. Clearly, a commutative semigroup is max-min type if and only if it becomes a “max" semigroup via : for m,n ∈ S,  m<n if and only if m· n=n ≠ m.

  (iii) Let (S,·) be a semigroup with identity e. For m,n ∈ S we usually write m· n = mn.


  (a) A non-empty subset T of S is called an ideal in S if TS ⊂ T and ST ⊂ T, where TS := {ts  : t ∈ T, s∈ S} and similarly for ST. As in <cit.>, an ideal T ( S) in S will be called a prime ideal if the complement S \ T of T in S is a semigroup.

  (b) Let E(S) denote the set of idempotent elements in S, i.e., the set of elements n ∈ S such that n^2=n. We write E_0(S)=E(S)\{e} and S= S \ E(S).

  (c) Let G(S) denote the set {g ∈ S: ∃  h ∈ S   with   gh=hg=e}. Then G(S) is a group contained in S called the maximal group. Note that G(S) ∩ E(S)= {e}.  Set G_1(S)= {g ∈ G(S):   gm=m  for all  m ∈ E_0(S)}. Clearly, G_1(S) is a subgroup of G(S).  Note that members of G_1(S) act on E_0(S) as the identity via left multiplication of (S, ·). 



   Let (S,·) be a semigroup with identity e. 
 
  (i)  We call (S, ·) inverse-free in case for m,n ∈ S, mn=e holds if and only if m=n=e. This condition is equivalent to saying that m or n is equal to e.
 
 
  (ii) (S, ·) is called action-free if G_1(S)= {e}. 


 


 
  (i) (ℤ_+,+ ), ((0,1], ×) and ([1, ∞ ), ×) are inverse-free semigroups which are not max-min type.
 
  (ii) (ℤ_+, ) and (ℤ_+ ∪{∞}, ) are max-min type inverse-free  semigroups. 
 
  (iii) A commuting  set of orthogonal projections on a Hilbert space ℋ containing the identity operator I_ℋ on ℋ, with respect to composition of two operators, form a semigroup with identity which is  inverse-free.
   
 We know that if (S, ·) is a commutative semigroup then (E(S), ·) is a semigroup. Interrelations are collected in the following proposition whose proof is straight-forward.
   Let (S,·) be a semigroup with identity e.
 
 
  (i) If (S,·) is inverse-free then G(S)= {e}. Converse part is also true if S is commutative.
 
  (ii) If (S, ·) is max-min type then (S,·) is inverse-free.
 
  (iii) Suppose S\{e} is non-empty. 
 
 
  (a) (S \{e}, ·) is a semigroup if and only if  (S,·) is inverse-free.
 
  (b) If (S, ·) is inverse-free then (S\{e}, ·) is an ideal in (S, ·).
 

 
  (iv)   Suppose E_0(S) is non-empty. If (S, ·) is inverse-free and commutative then (E_0(S), ·) is a semigroup.
 
  (v) For a commutative semigroup (S, ·),  S is an ideal if and only if S is a prime ideal.
  
  
 (Dichotomy).  Let S be a  semigroup. For m ∈ S either m^j,  j =1,2, … are all distinct or m^j is an idempotent for some j ∈ℕ.
 
 We will say m is of infinite order in the first case and of finite order in the second case.
 
 
   Here we provide an example of a commutative semigroup (S, ·), in which G(S)=G_1(S) ≠{e} and thus, S is not action-free. 
 
 We take S= (ℤ_+ ×{0}) ∪{(0,1)} and define `·' as follows: 
    (i,0) · (j,0)   =    (max{i,j},0)    for   i,j∈ℤ_+,
     (j,0) · (0,1)   =    (0,1)· (j,0)=(j,0)  for  j ≠ 0, 
      (0,0) · (0, 1)   =   (0,1)·(0,0) =(0,1) and
     (0,1) · (0, 1)   =   (0,0).

  Then e= (0,0) and E_0(S)= ℕ×{0}. Also G(S)= {(0,α): α =0,1 }=G_1(S).   
 
 The semigroup in our next example is action-free but not inverse-free.
   Let S= (ℤ_+, max) × ({0,1}, addition mod 2), a commutative semigroup with identity e= (0,0). It is easy to see that E_0(S)= ℕ×{0}, G(S)= {(0, α): α = 0,1}, and G_1(S)= {e}.
 
 In this paper, we shall mainly be concerned with semigroups equipped with the discrete topology, in short, discrete semigroups.
  Let (S,<) be a linearly ordered set as in Item <ref> (i).


  (i) Let (S,<, max) be the “max" semigroup as in Item <ref> (i)(a) above. We equip S with the order topology τ_0 and assume that S has an identity e. Then τ_0 is discrete if and only if 

(a) e has an immediate successor, and, 

(b) each m≠ e has an immediate predecessor as well as an immediate successor.

  (ii)Two different types of examples for (i) above can be provided by 

(a) (ℤ_+, <, max) or its non-empty finite subsets, and, 

(b) {0}∪{m ±1/n+2: m, n ∈ℕ} considered as a subset of the real line with the usual order and topology.


 
 
 

 §.§  Basics of hypergroups

 Here we come to the basics of hypergroups. Dunkl <cit.>, Jewett <cit.> and Spector <cit.> independently created locally compact hypergroups (same as `convos' in <cit.>) under different names with the purpose of doing standard harmonic analysis. In this paper we are mostly concerned with commutative discrete semiconvos or hypergroups. It is convenient to write the definition in terms of a minimal number of axioms. For instance, see (<cit.>, <cit.>).
 
   
 Let K be a discrete space. Let M(K) be the space of complex-valued regular Borel measures on K. Let M_F(K) and M_p(K) denote the subset of M(K) consisting of measures with finite support and probability measures respectively. Let M_F,p(K)= M_F(K) ∩ M_p(K). At times, we do not distinguish between m and δ_m for any m ∈ K because m ↦δ_m is an embedding from K into M_p(K). Here δ_m is the unit point mass at m, i.e., the Dirac-delta measure at m. 
  
   We begin with a map * : K × K → M_F,p(K). Simple computations enable us to extend `*' to a bilinear map called convolution, denoted by `*' again, from M(K) × M(K) to M(K). At times, for certain n ∈ K we will write q_n for δ_n* δ_n and Q_n for its support.
   
    A bijective map ∨: m ↦m̌ from K to K is called an involution if m̌̌̌=m. We can extend it to M(K) in a natural way.  
   
   A pair (K,*) is called a  discrete semiconvo if the following conditions hold.
 
 
  * The map * : K × K → M_F,p(K) satisfies the associativity condition 
    (δ_m*δ_n)*δ_k = δ_m* (δ_n*δ_k)   for all  m, n, k ∈ K.

 
  * There exists (necessarily unique) element e ∈ K such that 
    δ_m*δ_e = δ_e*δ_m = δ_m     for all   m ∈ K.
 
 
 A discrete semiconvo (K,*) is called commutative if δ_m*δ_n= δ_n*δ_m for all m,n ∈ K.
 
  A triplet (K,*, ∨) is called a discrete hypergroup if 
 
 
  * (K,*) is a discrete semiconvo,
 
  * ∨ is an involution on K that satisfies 
 
  (i)  (δ_m*δ_n )̌= δ_ň*δ_m̌ for all m,n ∈ K and 
 
  (ii) e ∈(δ_m*δ_ň) if and only if m=n.
  
  
 A discrete hypergroup (K,*, ∨) is called hermitian if the involution on K is the identity map, i.e., m̌=m for all m ∈ K. 
 
  Note that a hermitian discrete hypergroup is commutative.
  
   We write (K,*) or (K,*,∨) as K only if no confusion can arise. 

 Let K be a commutative discrete hypergroup. For a complex-valued function χ defined on K,  we write χ̌(m):= χ(m̌) and χ(m*n) = ∫_K χ  d(δ_m*δ_n) for m,n ∈ K. Now, define two dual objects of K:
 
    𝒳_b(K)= {χ∈ℓ^∞(K): χ≠ 0, χ(m*n)= χ(m) χ(n)  for all   m,n ∈ K },
 
    K= {χ∈𝒳_b(K) : χ̌= χ,  χ(m̌)= χ(m)  for all m ∈ K  }.
 Each χ∈𝒳_b(K) is called a character and each χ∈K is called a symmetric character. With the topology of pointwise convergence, 𝒳_b(K) and K become compact Hausdorff spaces. In contrast to the group case, these two dual objects need not be the same and also need not have a hypergroup structure. 
 
 Now we give some examples of hypergroups.
 Polynomial hypergroups: This is a wide and important class of hermitian discrete hypergroups in which hypergroup structures are defined on ℤ_+. This class contains Chebyshev polynomial hypergroups of first kind, Chebyshev polynomial hypergroups of second kind, Jacobi hypergroups, Laguerre hypergroups etc. For more details see <cit.> and <cit.>. 

For Illustration, we describe CP, the Chebyshev polynomial hypergroup of first kind which arises from the Chebyshev polynomials of first kind. In fact, they define the following convolution `*' on ℤ_+:

    δ_m*δ_n= 1/2δ_|n-m|+1/2δ_n+m   for   m,n ∈ℤ_+.
  For any k ∈ℕ, K= k ℤ_+ with *|_K × K makes K a discrete hypergroup in its own right. 
 The Chebyshev polynomial hypergroup of second kind (ℤ_+, *) arises from the Chebyshev polynomials of second kind and the convolution '*' on ℤ_+ is given by 
    δ_m*δ_n= ∑_k=0^min{m ,n}|m-n|+2k+1/(m+1)(n+1)δ_|m-n|+2k.



  Let H_a={0,1,2, …, ∞}, 0<a ≤1/2, be the one-point compactification of ℤ_+. Dunkl and Ramirez <cit.> defined a convolution structure on H_a to make it a (hermitian) countable compact hypergroup. For a prime p, let Δ_p be the ring of p-adic integers and 𝒲 be its group of units, that is , {x=x_0+x_1p+ …+ x_np^n+ …∈Δ_p : x_j = 0,1, …,p-1  for  j ≥ 0  and  x_0 ≠ 0  }. For a=1/p, H_a derives its structure from 𝒲-orbits of action of 𝒲 on Δ_p by multiplication in Δ_p. 
 
Next, Dunkl and Ramirez make the symmetric dual space H_a of H_a into a hermitian discrete hypergroup. The members of H_a are given by {χ_n : n ∈ℤ_+}, where, for k ∈ H_a, 
    χ_n(k)=   0    if  k <n-1, 
    a/a-1   if  k=n-1, 
     1    if   k ≥ n     (or   k = ∞).

Then the convolution `*' on K=ℤ_+ identified with H_a={χ_n : n ∈ℤ_+} is dictated by pointwise product of functions in H_a, that is: 

    χ_m χ_n    =   χ_max{m,n}   for   m ≠ n, 
    χ_0^2   =   χ_0,     χ_1^2 = a/1-aχ_0+ 1-2a/1-aχ_1,
    χ_n^2   =   a^n/1-aχ_0+ ∑_k=1^n-1 a^n-kχ_k+1-2a/1-aχ_n      for  n ≥ 2.


   We call (K,*) a (discrete) Dunkl-Ramirez hypergroup.
   
    H_a has a good spectral synthesis in the sense that every closed subset of H_a is a set of spectral synthesis for the  Fourier algebra A(H_a) <cit.>. Chilana (now, Ajit Iqbal Singh) and Ajay Kumar <cit.> have strengthened this further.
  We note a few more standard facts about a Dunkl-Ramirez hypergroup (K,*). We follow the notation as in 2.1 and 2.2 above.


  (i) For n ∈ℤ_+, (ℒ_n ∪{n}, *) is a subhypergroup of (K,*).

  (ii) Let a ≠1/2. For n ∈ℤ_+,  Q_n= ℒ_n ∪{n} and as a consequence, for n ∈ℕ, #Q_n ≥ 2.

  (iii) Let a= 1/2. 


  (a) For n ∈ℕ,  Q_n= ℒ_n. For n ≥ 2, #Q_n ≥ 2 whereas for n=1,  #Q_n=1. 

  (b) In fact, (ℒ_1 ∪{1},*) is a group isomorphic to ({0,1}, addition mod  2) 


 

  
 
  

§ HYPERGROUPS ARISING FROM HYPERGROUP DEFORMATIONS OF IDEMPOTENT ELEMENTS OF “MAX" SEMIGROUPS
 
 To begin with, we assume that (S,·) is a commutative  discrete semigroup with identity e such that E_0(S) ≠ϕ. For q ∈ M_p(S),  let Q be its support, in short, (q) and q(j)= q({j}) for j ∈ S. Then Q is countable. Observe that q=∑_j ∈ Q q(j) δ_j with q(j)>0 for each j ∈ Q and ∑_j ∈ Q q(j)=1. We prefer this to the usual form ∑_j ∈ S q(j) δ_j with q(j) ≥ 0 for j ∈ S and ∑_j ∈ S q(j)=1, unless otherwise stated.
  A probability measure q on S with #Q ≥ 2 will be called non-Dirac.
 
 

 §.§ Motivation
 
 We note that a Dunkl-Ramirez hypergroup (as in Example <ref> above) is a hermitian (hence commutative)  discrete hypergroup K=H_a   (0<a≤1/2) and its convolution `*' arises as a hypergroup deformation of the semigroup (ℤ_+, ·), where m · n= {m, n} in the sense that δ_m*δ_n= δ_mn for m ≠ n or, m=n=0 and for m=n ≠ 0, we have
 
    δ_1*δ_1    =   a/1-aδ_0+1-2a/1-aδ_1,
    δ_n*δ_n    =   a^n/1-aδ_0+ ∑_k=1^n-1 a^n-kδ_k + 1-2a/1-aδ_n      for   n ≥ 2.


 Also, each element of ℤ_+ is an idempotent in (ℤ_+, ·). Further, for n ∈ℤ_+, (ℒ_n ∪{n}, *) arises as a hypergroup deformation of the  finite subsemigroup (ℒ_n ∪{n}, max) of (ℤ_+, ·). This motivates the construction of a  commutative discrete  semiconvo or discrete hypergroup structure on S for a commutative  discrete semigroup (S, ·) by deforming the product. We elaborate as follows.
 
 

 §.§ Hypergroups deformations of  “max" semigroups 
  We now confine our attention to  a “max" semigroup (S, <, ·) as in Item <ref>(i)(a). We assume that S has an identity e and it is equipped with the discrete topology.
 We try to deform this discrete semigroup (S, ·) into a  hermitian (hence commutative) discrete hypergroup by deforming the product on the diagonal of S \{e}. 


  §.§.§ Preparatory material
  (i) For a discrete space K and a complex-valued regular Borel or a non-negative measure μ on K, we usually write μ(j) for μ({j}).

(ii)(a) Let (K,*) be a discrete hermitian hypergroup. Then by <cit.> the Haar measure λ on K is given by: λ(e)=1 and for e ≠ n ∈ K, λ(n)= 1/(δ_n*δ_n)(e).  

 (b) Let K be a Dunkl-Ramirez hypergroup H_a (0< a ≤1/2). Then λ(n)= 1-a/a^n for all n ∈ℕ. Further, for n ∈ℕ,  we have  

    δ_n*δ_n(0)    =   a^n/1-a= 1/λ(n),
    δ_n*δ_n(k)   =    a^n-k =
    λ(k)/λ(n)  for  1≤ k <n   and
    δ_n*δ_n(n)   =   1-2a/1-a= 1- ∑_0 ≤ k<n (δ_n*δ_n)(k)= 1- (δ_n*δ_n)(ℒ_n)= λ(n)-λ(ℒ_n)/λ(n).
 
(c) (δ_n*δ_n)(n)=0 for some n ∈ℕ if and only if a=1/2 if and only if (δ_n*δ_n)(n)=0 for all n ∈ℕ. In this case, λ(n)=2^n-1 for n ∈ℕ. 






We try to replace (ℤ_+,<, ·) in the Dunkl-Ramirez hypergroups by a discrete “max" semigroup (S,<, max) with identity in our next theorem.

We give a set of necessary and sufficient conditions for (S,*) to become a  hermitian discrete hypergroup with convolution product `*' defined as follows: 
 
    δ_m*δ_n = δ_n*δ_m    =   δ_m· n (= δ_max{m,n})       m,n ∈ S with  m ≠ n,  or,  m=n=e,
    δ_n*δ_n    =    q_n            for  n ∈ S\{e}.

  Here, q_n is a probability measure on S with finite support Q_n containing e and has the form ∑_j ∈ Q_n q_n(j) δ_j with q_n(j)>0 for j ∈ Q_n and ∑_j ∈ Q_n q_n(j)=1.
  
   This is equivalent to looking for conditions on S, Q_n's, an S-tuple {v_n}_n ∈ S in [1, ∞) with v_e=1 and v_n= 1/q_n(e) for n ∈ S \{e}  and q_n(j) for j ∈ Q_n \{e},  n ∈ S \{e}.
  
  
  
   Let (S,<, ·) be a discrete (commutative) “max" semigroup with identity e and ` *' and other related symbols as above. Then (S,*) is a hermitian discrete hypergroup if and only if the following conditions hold. 
 
 
  (i) Either S is finite or (S,<,·) is isomorphic to (ℤ_+,<, max).
 
  (ii) For n ∈ S  \{e}, we have ℒ_n ⊂ Q_n ⊂ℒ_n ∪{n}. 
 
  (iii)  If #S > 2, then for e ≠ m <n in S, we have 
 
 
  (a) q_n(e)=q_n(m)q_m(e) and 
 
  (b) q_n(e) ( 1+ ∑_e ≠ k ∈ℒ_n1/q_k(e)) ≤ 1; 
  
 or, equivalently, with v_n= 1/q_n(e) for n ∈ S,
 
  (iii)'  If #S > 2, then for e ≠ m <n in S, we have 
 
 
  (a)q_n(m)= v_m/v_n and 
 
  (b) ∑_k ∈ℒ_nv_k ≤ v_n. 
   
 
Suppose (S,*) is a hermitian discrete hypergroup. 

We prove (i) in two steps.

Step (α). Let n ∈ S \{e} and m<n . Associativity of (S, *) demands that (δ_n*δ_n)*δ_m=δ_n*(δ_n*δ_m). This gives that q_n*δ_m=q_n. But e ∈ Q_n; therefore, m ∈ Q_n. Hence ℒ_n ⊂ Q_n. Since #Q_n < ∞ we have #ℒ_n < ∞.  Clearly, ℒ_e= ϕ is finite.

Step (β). Assume that S is not finite. Consider any n ∈ S. By Step (α) ℒ_n ∪{n} is finite and, therefore,  𝒰_n = S \ (ℒ_n ∪{n}) is non-empty. Choose any m ∈𝒰_n.  Suppose that m is not immediate successor of n.  Then, there exists an element m' ∈ S such that n<m'<m. So 𝒰_n ∩ℒ_m is finite and non-empty because m' ∈𝒰_n ∩ℒ_m and ℒ_m is finite by Step (α). Since 𝒰_n ∩ℒ_m is finite we can enumerate its elements in  order, say, n_1<n_2< ··· <n_s.  Choose the immediate successor n_1 of n. Thus each element n of S has an immediate successor, say, s(n). Now, start with x_0=e,
take x_1= s(e),  x_2=s(x_1), ···, x_j=s(x_j-1) and so on.  
Set W={x_j:  j ∈ℤ_+}. We claim that W=S. Suppose not; then there exists t ∈ S\ W. Therefore, t ≠ e, and hence t>e. But x_1 is immediate successor of e so t>x_1. Again, x_2 is the immediate successor of x_1, so t>x_2. Repeating this process, we get t>x_j for every j ∈ℤ_+. Hence W ⊂ℒ_t. This shows that W is finite because ℒ_t is finite by Step (α). This gives a contradiction. So, S=W. 

 Therefore, either S is finite or (S,<,·) is isomorphic to (ℤ_+,<, max).

(ii) Consider any n ∈ S \{e}. Let, if possible,  Q_n ⊄ℒ_n ∪{n}. Then there exists m >n with m ∈ Q_n. The associativity of `*' demands that (δ_n*δ_n)*δ_m= δ_n*(δ_n*δ_m),  which in turn gives that q_n*δ_m=δ_m. Now, L.H.S. = q_n(m) δ_m*δ_m + ∑_m ≠ j ∈ Q_n q_n(j) δ_jm=q_n(m) q_m + ∑_m ≠ j ∈ Q_n q_n(j) δ_jm. By Step (i) (α),  ℒ_m ⊂ Q_m. Now, Q_m ⊂(L.H.S.)=(R.H.S.)= {m}. But n ∈ℒ_m. Thus, we have n=m, which is a contradiction. Therefore, Q_n ⊂ℒ_n ∪{n}. Hence, using Step (i)(α), we get ℒ_n ⊂ Q_n ⊂ℒ_n ∪{n} for n ∈ S \{e}.  



(iii) Suppose #S>2 and e ≠ m <n in S. 


  (a) As in the proof of Step(i)(α), 
 q_n*δ_m= q_n. Now, as by (ii), m ∈ Q_n, we get 

    =     q_n(e) δ_m + ∑_e ≠ j ≠ mj ∈ Q_n q_n(j) δ_jm + q_n(m) δ_m*δ_m 
       =    q_n(e) δ_m + ∑_e ≠ j ≠ mj ∈ Q_n q_n(j) δ_jm + q_n(m) ∑_j' ∈ Q_m q_m(j') δ_j'  and
 

    = q_n(e) δ_e + ∑_e ≠ k ≠ m,   k ∈ Q_n q_n(k) δ_k + q_n(m) δ_m.

Since jm ≠ e for j,m ∈ S \{e} and R.H.S. = L.H.S., we get q_n(e)= q_n(m)  q_m(e). 


  (b) 
 Because n ∈ S \{e}, we get 0< ∑_ j ∈ℒ_n q_n(j) ≤∑_j ∈ Q_n q_n(j) =1. By (iii)(a), q_n(j)= q_n(e)/q_j(e) for all j ∈ℒ_n \{e}. Therefore, we get 
    q_n(e)( 1+ ∑_e ≠ j ∈ℒ_n1/q_j(e)) = q_n(e) +∑_e≠ j ∈ℒ_nq_n(e)/q_j(e)= ∑_j∈ℒ_n q_n(j) ≤ 1.


It can be seen easily that (iii)' is only a rewording of (iii).

  
For the  converse part, assume that (i)-(iii) all hold. We note right at the outset that, because condition (ii) is satisfied, for n ∈ S\{e}, we may write q_n=δ_n*δ_n= ∑_j ∈ℒ_n ∪{n} q_n(j) δ_j= ∑_j ≤ n q_n(j) δ_j with q_n(j) >0 for j ∈ℒ_n and q_n(n) ≥ 0. 

 To see that (S,*) is a hermitian  discrete hypergroup, first note that the identity element e of semigroup (S, ·) works as the identity of (S,*), i.e., δ_n*δ_e= δ_e*δ_n=δ_n for every n ∈ S. 

Next, we show that e ∈(δ_m*δ_n) if and only if m=n. It is trivial in case m=n=e.  If m =n ≠ e then e ∈(δ_n*δ_n)= Q_n. Now, assume that e ∈(δ_m*δ_n). Let, if possible, m ≠ n, so δ_m*δ_n= δ_mn. Without loss of generality we can assume that m<n, so δ_mn=δ_n and n>e. Therefore, e ∈(δ_m*δ_n) shows that n =e, which is a contradiction. Hence m=n.

Now, we prove the associativity condition of (S,*), i.e., for m,n,k ∈ S

    (δ_m*δ_n)*δ_k= δ_m*(δ_n*δ_k).

If any one or more of {m,n,k} are e then (<ref>) is trivial. We now consider the cases when none of m,n and k are e. 


  Case (i) If m ≠ n ≠ k ≠ m, we first consider the subcase m<n<k. So, L.H.S. of (<ref>) = δ_n*δ_k =δ_k and R.H.S. of (<ref>) = δ_m*δ_k=δ_k. Therefore, L.H.S. =R.H.S. Other subcases can be dealt with in a similar way and hence (<ref>) holds. 

 
  Case (ii) Out of m,n,k exactly two are distinct. We get #S>2.


  (α) We consider first the following form of (<ref>):  (δ_n*δ_n)*δ_m= δ_n*(δ_n*δ_m) with e ≠ n ≠ m ≠ e, which is the same as 
 
    q_n*δ_m = δ_n*δ_nm

 When n<m, R.H.S. of (<ref>) is equal to δ_n*δ_m=δ_m. The condition that Q_n ⊂ℒ_n ∪{n} implies that Q_n· m ={m} and using this, we get  L.H.S. of (<ref>) = ∑_j ∈ Q_n q_n(j) (δ_j*δ_m)= ∑_j ∈ Q_n q_n(j) δ_jm=( ∑_j ∈ Q_n q_n(j) )δ_m =δ_m as ∑_j ∈ Q_n q_n(j) =1 . Therefore, L.H.S.= R.H.S. and hence (<ref>) holds.

We now come to the case when m<n. Then R.H.S. of (<ref>) is q_n= ∑_j ≤ n q_n(j) δ_j with q_n(j)>0 for j <n and q_n(n) ≥ 0. 
  So L.H.S. of (<ref>) 

    =   ∑_j ≤ n q_n(j) (δ_j*δ_m)= ∑_j ≠ mj≤ n q_n(j) δ_jm+ q_n(m) ∑_k ≤ m q_m(k) δ_k 
       =   ∑_m<j ≤ n  q_n(j) δ_j + ( ∑_j <m q_n(j) ) δ_m + ∑_k<m q_n(m) q_m(k) δ_k + q_n(m) q_m(m)δ_m .


 
By (iii)(a) q_n(k)  q_k(e)= q_n(e) for k<n. So we get q_n(m)  q_m(j)= q_n(j) for all j <m. Therefore, L.H.S. of (<ref>)

    =   ∑_m<j ≤ n q_n(j) δ_j + ( ∑_j<m q_n(m) q_m(j) +  q_n(m) q_m(m)) δ_m+  ∑_k <m q_n(k) δ_k 
       =   ∑_m<j ≤ n  q_n(j) δ_j +  q_n(m) ( ∑_j ≤ m q_m(j)  ) δ_m+  ∑_k<m q_n(k) δ_k.

Further, we have ∑_j ≤ m q_m(j) =1. So,
L.H.S. of (<ref>) =  ∑_k ≤ n q_n(k) δ_k = q_n.

Therefore, we get L.H.S. of (<ref>) = R. H.S. of (<ref>) and hence (<ref>) holds.

  (β) Now, consider the following form of (<ref>) with m<n, (δ_n*δ_m)*δ_n= δ_n*(δ_m*δ_n). R.H.S. = δ_n*δ_n = (δ_n*δ_m)*δ_n= L.H.S. 

  (γ) The case m>n of the form (<ref>) as in (β) above follows simply because L.H.S. = δ_m*δ_n= δ_m= R.H.S.



  (iii) If m=n=k then (<ref>) follows from the fact δ_s*δ_n= δ_n*δ_s for s ∈ Q_n.

Hence (<ref>) holds and consequently (S,*) is a hermitian discrete hypergroup.


  In view of condition (i) in Theorem <ref> and Remark <ref>(i) we note that for “max" semigroups (S,<, max) of interest to us the discrete topology and the order topology coincide (cf. <cit.>, <cit.>).
 

We now interpret conditions (i)-(iii) in Theorem <ref> above. Because a finite “max" semigroup is isomorphic to (ℤ_k= {j: 0 ≤ j < k}, <, max) for some k ∈ℕ, we confine our attention to ℤ_+ to begin with. 

Let 𝒱 be the set of sequences v=(v_j)_j ∈ℤ_+ in [1, ∞) which satisfy (i) v_0=1 and (ii) v_n ≥∑_j ∈ℒ_nv_j for n ∈ℕ. 

For n ∈ℕ, let u_n= v_n- ∑_j ∈ℒ_nv_j. Then simple calculations give the following:
 
    v_0    =    1 , 
     v_1   =   1+u_1,
     v_2   =   (2+u_1)+u_2, 
     v_3    =    (2^2+2u_1+u_2)+u_3, 
    ⋮   =   ⋮, 
    to elaborate,
     v_n    =    (2^n-1+2^n-2u_1+ …+u_n-1)+u_n     for  n ≥ 3.
 

Alternatively, we may start with a sequence u=(u_n)_n ∈ℕ in [0, ∞) and define (v_n)_n ∈ℤ_+ as above. Let 𝒰 be the set of such sequences (u_n)_n ∈ℕ's. The sets 𝒰 and 𝒱 have cardinality 𝔠. 

 The same is true if we confine our attention to domain ℤ_k= {j: 0 ≤ j < k} for k ∈ℕ, instead of ℤ_+.
 
 
 
   For each v ∈𝒱 (or the corresponding u ∈𝒰), there is one and only one (hermitian) hypergroup deformation (ℤ_+,*) of (ℤ_+,<, max) which satisfies (δ_n*δ_n)(0)= 1/v_n, n ∈ℤ_+. Further, for this deformation, the convolution `*' and the Haar measure λ satisfy the following conditions.
  
 
  (i) λ(n)=v_n for n ∈ℤ_+ and  λ(n)-λ(ℒ_n)=u_n for n ∈ℕ.
 
  (ii) λ(ℒ_n) ≤λ(n) for n ∈ℕ.
 
  (iii) For n ∈ℕ, 
 
 
  (a) δ_n*δ_n(m)= λ(m)/λ(n) for m<n,
 
  (b) δ_n*δ_n(n)=λ(n)-λ(ℒ_n)/λ(n),
 
  (c) δ_n*δ_n(m)=0 for m>n.
 
 
  (iv) For n ∈ℕ,
(δ_n*δ_n) = ℒ_n    if  λ(n)=λ(ℒ_n), 
ℒ_n ∪{n}   if  λ(n)>λ(ℒ_n).
 
  
 
 (i) This is immediate from <ref>(ii)(a) and expressions for (v_n)_n∈ℤ_+ and (u_n)_n ∈ℕ given above.
 
 (ii) This follows from Theorem <ref>(iii)(b).
 
 (iii) We have only to appeal to Theorem <ref>(iii)(a).
 
 (iv) This follows from (iii).
 
 
  We compare Theorem <ref> and Corollary <ref> above with the Dunkl-Ramirez hypergroups (Example <ref> above). We freely use Remark <ref>, Section <ref> and Item <ref> above.
  

  (a) For the Dunkl-Ramirez hypergroup H_a (0< a ≤1/2), q_n(0)= a^n/1-a,   n ∈ℕ. Further, Q_m= ℒ_m for some m ∈ℕ if and only if a = 1/2 if and only if Q_n=ℒ_n for all n ∈ℕ.

  (b) For our theorem, for any arbitrarily fixed m ∈ S \{e},Q_m=ℒ_m  if and only if q_m(e) (1+∑_e≠ j ∈ℒ_m1/q_j(e))=1. Under the correspondence set up after Theorem <ref>, the latter condition  can be shortened to u_m =0. 

  (c) Suppose #S>2. For any strictly increasing sequence (n_j) in S \{e} contained in S, considered as a subsemigroup of (ℤ_+,<, max) in view of Theorem <ref> (i) above, we can find (q_n(e))_n ∈ S leading to a deformation satisfying Q_n_j= ℒ_n_j ∀ j and Q_n= ℒ_n ∪{n} for n ∈ S \{e} but not equal to any n_j. 

 


 §.§ The dual objects of (S,*)
 
 
We now come to the dual objects 𝒳_b(S) and S of hypergroup (S,*), as in Theorem <ref> and Corollary <ref> above. For this purpose, to begin with, we fix any v ∈𝒱 (or the corresponding u ∈𝒰) and consider the corresponding deformation (ℤ_+,*) as obtained in Theorem <ref> and Corollary <ref>. We freely follow the concepts, results and notation set up in this process. 




  Let (S,<, ·) ≅ (ℤ_+, <, max) and other symbols  satisfy the conditions (ii)-(iii) of Theorem <ref> and let (S,*) be the corresponding deformed hypergroup with the Haar measure λ. Then the dual objects  𝒳_b(S) and S of (S,*) are equal. Equipped with the topology of uniform convergence on compact subsets of S, S can be identified with the one point compactification ℤ_+^* (= ℤ_+ ∪{∞}) of ℤ_+.  More precisely, the identification is given by k ↦_k, where _∞(n) =1 for all n ∈ℤ_+, and, for k ∈ℤ_+, _k is given by 

    _k(n) = 1    if   n ≤ k, 
    β_k    if   n=k+1, 
      0    if  n>k+1,
 
where, β_k= -λ(ℒ_k+1)/λ(k+1)= -∑_j ∈ℒ_k+1v_j/v_k+1 =u_k+1/v_k+1-1= δ_k+1*δ_k+1(k+1)-1=q_k+1(k+1)-1.
 
 
  Before we start the proof, it is helpful to give a diagrammatic interpretation (Table <ref>) of the functions _k's, k ∈ℤ_+. As in the proof of Theorem <ref>, for n ∈ℕ, we may write q_n= ∑_j=0^n q_n(j) δ_j with q_n(j)>0 for 0 ≤ j<n, but q_n(n) ≥ 0 and ∑_j=0^n q_n(j)=1.
 
 We note that, for k ∈ℤ_+, β_k=q_k+1(k+1)-1 and -1 ≤β_k <0.
 
 
  
    

 First, we check that for k ∈ℤ_+^*=ℤ_+ ∪{∞}, _k  is a character. Clearly, _∞ is a character. Fix k ∈ℤ_+. It is easy to see that _k(m*n)= _k(m) _k(n) for all m ≠ n ∈ℤ_+. 
 
 It remains to check that for m ∈ℤ_+,  (_k(m))^2= _k(m*m), i.e.,
 
    (_k(m))^2=  ∑_j=0^m q_m(j) _k(j).

 Let m ∈ℤ_+.
 
 
  Case (i) m ≤ k : We have _k(m)= 1 and on the other hand, _k(m*m)= ∑_j=0^m q_m(j) _k(j) = ∑_j=1^m q_m(j) =1. Therefore, (<ref>) holds.
 
 
  Case (ii) m=k+1 : We have 
    _k(m*m)   =   ∑_j=0^k+1 q_k+1(j) _k(j)= ∑_j=0^k q_k+1(j)+ q_k+1(k+1) (q_k+1(k+1)-1)
       =    (1-q_k+1(k+1))+(q_k+1(k+1))^2- q_k+1(k+1) 
       =    (q_k+1(k+1)-1)^2 = (_k (m))^2.

 Therefore, (<ref>) holds.
 
  Case (iii) m>k+1 : We have _k(n)=0 for n>k+1. 
  So, _k(m*m)= ∑_j=0^k q_m(j)+ q_m(k+1)(q_k+1(k+1)-1).
 Since by Theorem <ref> and by proof of part (iii)(b) of Theorem <ref>, we know that for 1 ≤ j < m, q_m(j)= q_m(0)/q_j(0) and 1-q_k+1(k+1) = q_k+1(0) ( 1+ ∑_j=1^k 1/q_j(0)), we get 
 
    _k(m*m)= q_m(0) (1+∑_j=1^k 
     1/q_j(0) +1/q_k+1(0)(q_k+1(k+1)-1) )=0 = (_k(m))^2.

 
 Therefore, _k is a character. Also, _k is real-valued and S is hermitian. So, _k is a symmetric character.
 
  Now, note that for k_1 ≠ k_2 ∈ℤ_+^*, we have _k_1≠_k_2. We will show that injective map k ↦_k from ℤ_+^* to 𝒳_b(S) is onto. 
  
  To prove this take any χ∈𝒳_b(S) such that χ≢1. Since χ(0)=1, there exists a least s ∈ℕ such that χ(s) ≠ 1. Let m,n ∈ℤ_+ with m<n. We have 
  
    χ(m) χ(n) = χ(m*n)= χ(n).

  This shows that either χ(m)=1 or χ(n)=0. As a consequence, χ(n)=0 for all n>s.
  
  Now consider the set 
    A={j ∈ℕ: χ(t)=0  for all  t ≥ j }.

  Then s+1 ∈ A and so A ≠ϕ. Hence, there exists a least element, say, w in A. 
   
   Let, if possible, w=1. Then χ(0)=1 and χ(t) =0 for all t ≥ 1. So, for any m ≥ 1, we get χ(m*m)= χ(m)χ(m)=0. On the other hand, we have χ(m*m)=  ∑_j=0^m q_m(j) χ(j)= q_m(0) >0, which is a contradiction. Therefore, w>1. 
   
   Note that χ(n)=1 for all n < w-1. In fact, let,  if possible, χ(n_1) ≠ 1 for some n_1<w-1. Therefore we get χ(t)=0 for t ≥ w-1, which shows that w-1 ∈ A. This is a contradiction to the fact  w is the least element of A. 
   
  Now, we come to χ(w-1)= β (say). Then, we have 
  
    β^2 = (χ(w-1))^2    =   χ((w-1)*(w-1))= ∑_j=0^w-1 q_w-1(j) χ(j)
        =   ∑_j=0^w-2 q_w-1(j) + q_w-1(w-1)  β.
   
This implies that β^2- q_w-1(w-1) β - (1-q_w-1(w-1))=0  and therefore, β = 1 or q_w-1(w-1)-1=β_w-2 where w-2≥ 0. 

Let, if possible, β =1. As w ∈ A, χ(w*w)= χ(w)χ(w)=0;  on the other hand, we have χ(w*w) = ∑_j=0^w q_w(j) χ(j) =∑_j=0^w-1 q_w(j) ≥ q_w(0) >0.  Hence, β= β_w-2.

Therefore, the character χ is given by 

    χ(n) =  1    if   n ≤ w-2,
    β_w-2   if   n=w-1, 
     0    if  n>w-1,

i.e., χ= _w-2.



 Therefore, the map k ↦_k is onto. As an immediate consequence, 𝒳_b(S) = S. 

Further, it is clear from Table <ref> that S equipped with the topology of uniform convergence on compact subsets of S (i.e.,  the topology of pointwise convergence in our case) is the one point compactification ℤ_+^* of ℤ_+.
  
  
    (i) Consider any countable infinite compact Hausdorff space X. Then any  probability measure μ on X has the form μ =  ∑_x ∈ Xμ(x) δ_x=∑_x ∈ Xγ_x δ_x (say), where γ_x ≥ 0 and ∑_x ∈ Xγ_x =1. Trivially, μ has compact support. 
In particular, if X=S as in Theorem <ref> above, μ = ∑_j ∈ℤ_+_j δ_j+ _∞δ_∞, where _j ≥ 0  ∀ j ∈ℤ_+^* and ∑_j ∈ℤ_+_j+_∞ =1.  

Let C= {  j ∈ℤ_+^*: γ_j >0}. Then the support of μ is given by  
    C=   C    if  C  is finite or ∞∈ C, 
     C ∪{∞}   otherwise.


(ii) For a non-constant complex-valued function f on S≅ℤ_+^*, which is eventually 1 in the sense that f(j)=1 for j≥ j_0 for some j_0 ∈ℕ,  we have, for  μ on S as in (i) above, 
    ∫_S f dμ = ∑_j ∈ℤ_+j<j_0f(j) _j+( 1 - ∑_j ∈ℤ_+j<j_0_j )= 1+  ∑_j ∈ℤ_+j<j_0 (f(j)-1) _j.
 
  
(iii) For concepts, notation and details related to our next theorem,  we refer to <cit.>, <cit.> and <cit.>. For the sake of convenience, we state Proposition 2.4.2 of <cit.> in the form we need as follows.
     
    “ S can be made into a hypergroup (with respect to pointwise multiplication) if for χ, χ' ∈S, there exists a (regular) probability measure μ_χ,χ'=∑_j ∈ℤ_+μ_χ, χ'(j) δ_χ_j+ μ_χ,χ'(∞) δ_χ_∞ = ∑_j ∈ℤ_+_j^χ, χ'δ_χ_j+ _∞^χ,χ'δ_χ_∞ (say) which satisfies 
    
  (a) for v ∈ℤ_+, 
    χ(v) χ'(v) = ∑_j ∈ℤ_+_j^χ, χ'_j(v)+_∞^χ,χ',
 
    
  (b) the map (χ, χ') ↦(μ_χ,χ') is continuous on ℤ_+^* ×ℤ_+^* → the space 𝒞(ℤ_+^*) of compact subsets of ℤ_+^* with Michael topology,
    
  (c) _∞∈(μ_χ, χ') if and only if χ=χ'.” 
    
    In this case, we may set δ_χ*δ_χ'= μ_χ,χ', take involution as identity and make S a hermitian hypergroup, in fact.
     




  Let (S,<, ·) ≅ (ℤ_+, <, max) and other symbols  satisfy the conditions (ii)-(iii) of Theorem 3.2 and let (S,*) be the corresponding deformed hypergroup with the Haar measure λ. The dual space S of (S,*) becomes a countable compact hermitian  hypergroup with respect to pointwise multiplication. More precisely, the convolution `*' on S is given by 
    δ__m*δ__n = δ__min{m,n}   for  m, n ∈ℤ_+  with    m ≠ n  or  m=n=∞, 
    ∑_j ∈ℤ_+_j^m δ__j         otherwise,

where, _j^m= 0 for j<m, _m^m= 1+β_m ≥ 0, and for p ≥ 1,
 _m+p^m= ∏_j=m^m+p-1-β_j/1-β_j+1>0, and, β_j's are as in Theorem <ref>.
Further, we also have 𝒳_b(S)=S≅ (S,*). 


We will freely use Remark <ref>.

It is clear from Table <ref> that δ_χ_m*δ_χ_n := δ__min{m,n} for m, n ∈ℤ_+^* with m ≠ n or m=n= ∞.

We now look for a suitable probability measure μ_m,m, in short, μ_m, for m ∈ℤ_+. Fix m ∈ℤ_+. By  Remark (<ref>)(i) above, we may write μ_m =∑_j ∈ℤ_+_j^m δ_j+ _∞^m δ_∞, where _j^m ≥ 0 for j ∈ℤ_+^* and ∑_j ∈ℤ_+_j^m + _∞^m =1. Now, for any t ∈ℕ, consider the function e_t on S defined by e_t(_n)= _n(t) for n ∈ℤ_+^*. We note from Table <ref> and Remark <ref>(ii) that the condition in  Remark <ref> (iii)(a) takes the form

    _m(t)^2-1 = ∑_n<tn ∈ℤ_+ (_n(t)-1) _n^m       for all  t ∈ℕ.
 
We may express it as:
 
    ([ 1-β_0     0     0     ⋯;     1 1-β_1     0     ⋯;     1     1 1-β_2     ⋯;     ⋮     ⋮     ⋮     ⋱ ]) ([ _0^m; _1^m; _2^m;    ⋮ ]) = ([ 1-_m(1)^2; 1-_m(2)^2; 1-_m(3)^2;         ⋮ ]).

  We consider any principal truncated submatrix of the infinite matrix on the L.H.S. of (<ref>) above. It is a lower triangular matrix and its diagonal entries 1-β_j's are all non-zero, and, therefore, it is invertible.  
 Now, for j ∈ℕ with j≤ m, 1-_m(j)^2=0. So we get ^m_j=0 for j<m . So the system (<ref>) can be replaced by the system:
 
    ([   1-β_m       0       0       ⋯;       1 1-β_m+1       0       ⋯;       1       1 1-β_m+2       ⋯;       ⋮       ⋮       ⋮       ⋱ ]) ([   _m^m; _m+1^m; _m+2^m;      ⋮ ]) = ([ 1-β_m^2;       1;       1;       ⋮ ])
 
 and _j^m=0 for j<m, if any.
  
  The first two rows of (<ref>) readily give, 
 
 
    _m^m= 1+β_m  ≥ 0,  and  _m+1^m= -β_m/1-β_m+1.
 
 
 We note that by Theorem <ref>, _m^m>0 if and only if δ_m+1*δ_m+1(m+1)>0.
 
 Consecutive pairs of rows then give that, for p ≥ 2, 
 
    _m+p^m = -β_m+p-1/1-β_m+p_m+p-1^m .

 
 So, for p ≥ 1,
 
    _m+p^m = ∏_j=m^m+p-1-β_j/1-β_j+1>0.

 



Let s_p= ∑_j=0^p _m+j^m  for p ≥ 1. First note that 
    s_1= _m^m+_m+1^m= 1+ β_m-β_m/1-β_m+1= 1+ (-β_m/1-β_m+1) β_m+1= 1+_m+1^m β_m+1.

Assume that s_p= 1+_m+p^m β_m+p. Then 

    s_p+1   =    s_p+ _m+p+1^m=1+ _m+p^m β_m+p+ _m+p+1^m 
       =    1+ _m+p^m β_m+p- _m+p^m β_m+p/1-β_m+p+1
       =     1+ _m+p^m( -β_m+p/1-β_m+p+1) β_m+p+1 = 1+_m+p+1^m β_m+p+1.
 
Therefore, by mathematical induction we get s_p= 1+ _m+p^m β_m+p for p ≥ 1.

Let p ≥ 1.  Then 
    _m+p^m  β_m+p= β_m ∏_j=1^p-β_m+j/1-β_m+j,




which, by using the fact that -1 ≤β_k <0 for all k ∈ℤ_+, gives 
    0<-_m+p^m  β_m+p≤( 1/2)^p.

Therefore, ∑_j ∈ℤ_+_j^m = lim_p →∞ s_p = 1. So, _∞^m= 1- ∑_j ∈ℤ_+_j^m = 0.
 
  Hence, we may set μ_m = δ__m*δ__m = ∑_j ∈ℤ_+_j^m δ__j. 
 
 The continuity of the map (_m, _n) ↦(δ__m*δ__n)  is easy to  see by  noting that for m ≠ n or m=n= ∞, (δ__m*δ__n) is singleton {_min{m,n}} and for m ∈ℤ_+, (δ__m*δ__m) = {_k ∈S: k ≥ m }∪{_∞} or {_k ∈S: k > m }∪{_∞} according as δ_m+1*δ_m+1(m+1)>0 or δ_m+1*δ_m+1(m+1)=0.  
 
It can be readily checked that _∞ is the identity element. We see that _∞∈(δ__m*δ__n) if and only if m=n. In view of Remark (<ref>)(iii),  S is a compact hermitian hypergroup.




For the rest of the proof, it is convenient to index the elements of  S by ℤ_+^* under the identification k ↦_k.

Now, we calculate the dual space 𝒳_b(S), the space of continuous characters on S. We know that S ⊂S via n ↦_n, defined as _n(_k)= _n(k):= _k(n) for all k ∈ℤ_+^*. We can figure them out from Table <ref>; to elaborate, _0 ≡ 1, _1(0)=β_0, _1(k)=1 for k ≥ 1, and, for n≥ 2,  

    _n(k)=  0    if  k<n-1, 
    β_n-1   if  k=n-1, 
     1    if   k ≥ n.


Next, let ξ∈𝒳_b(S) such that ξ≢1. Consider the set 

    B= { s ∈ℤ_+: ξ(s) ≠ 1 }.
 

Then B ≠ϕ. Take any s ∈ B, so ξ(s)≠ 1. Note that for m ∈ℤ_+, n ∈ℤ_+^*  with m<n, we have ξ(m)= ξ(m)ξ(n), and therefore,  either ξ(m)=0 or ξ(n)=1. In particular, we get that for all  t ∈ℤ_+ with t <s, ξ(t)=0  and hence t ∈ B. 
This implies that ℒ_s ∪{s}⊂ B. So either B= ℤ_+ or B has a maximum element. 

Let, if possible, B= ℤ_+. Then, for each n ∈ℕ, ξ=0 on ℒ_n. Since this holds for every n ∈ℕ we get ξ≡ 0 on ℤ_+. Since ξ(∞)=1, we get that ξ is not continuous, a contradiction. Hence, B has a maximum element, say w.  Therefore, it follows from the definition of w that ξ(k)= 1 for all k>w. Also, since ξ(w) ≠ 1 we get ξ(k)=0 for k <w. 

Now, we calculate the value of ξ(w) = β (say). We have 
    β^2= ξ(w)^2= ξ(w*w)= _w^w β+∑_j>wj ∈ℤ_+_j^w.
 
This implies that β^2- _w^w β -(1-_w^w)=0 and therefore, β=1 or _w^w-1. But we know that β = ξ(w) ≠ 1 and hence ξ(w)= _w^w-1=β_w.
Now, set n=w+1 ≥ 1, then ξ= ξ_n.

 Hence (S,*) ≅𝒳_b(S).   
 
  (i)(a) The last part of Theorem <ref> can be proved directly with the help of <cit.>. Our proof is easy and self contained.
 
 (b) γ's in Theorem <ref> are related via _j^m= _j^0/_m^0 for j>m>0.
 
 (ii)(a) Any hypergroup (S,*) as in Theorem <ref> above may be thought of as hypergroup deformation of the compact linearly ordered semigroup (ℤ_+^*, <, min)  with the order topology (cf. <cit.>, <cit.>, <cit.>) on the diagonal of ℤ_+ ×ℤ_+.
 
 (b) The class of  hypergroups S as in Theorem <ref> above lies between the class of countable compact hypergroups H_a (see Section 2 above) of <cit.> and  the class of compact almost discrete hypergroups of <cit.>.
 
 (iii) Let (S,<,max) be a finite “max" semigroup of cardinality n. Then it can be identified with (ℤ_n, <, max).  Suitably truncated table of Table <ref> and truncated submatrices in (<ref>) and (<ref>) can be used to formulate and prove suitable analogues of Theorem <ref> and Theorem <ref>. 
 
  
  
  
  

§ SEMICONVOS OR HYPERGROUPS ARISING FROM DEFORMATIONS OF COMMUTATIVE DISCRETE SEMIGROUPS

We consider  a commutative discrete semigroup (S,·)  with identity e such that E_0(S) is not empty. We try to make (S,·) into a commutative discrete semiconvo or discrete hypergroup (S,*) by deforming the product on 𝒟_E_0(S):={(m,m): m ∈ E_0(S)}, the diagonal of E_0(S), or the idempotent diagonal of S, say.
 
 For n ∈ E(S), let q_n be a probability measure on S with finite support Q_n containing e. We express q_n= ∑_j ∈ Q_n q_n(j) δ_j with q_n(j)>0 for j ∈ Q_n and ∑_j ∈ Q_n q_n(j)=1. We look for necessary and sufficient conditions on S and { q_n: n ∈ E_0(S)} such that (S, *) with `*' defined below, is a commutative discrete semiconvo:
 
    δ_m*δ_n =  δ_n*δ_m     =    δ_mn       (m,n) ∈ S× S \𝒟_E_0(S),   
    δ_n*δ_n    =    q_n        n ∈ E_0(S).

 
 
It is convenient to consider various relationships among the properties of (S,·) and of (S,*) presented in the following theorems before coming to our main Theorem <ref>.
  Let (S,·) be a commutative discrete semigroup with identity e. Suppose (S,*) is a commutative discrete semiconvo, where `*' and other related notation and concepts are as above. Then the following are equivalent. 


  (i)  For n ∈ E_0(S),  Q_n ⊂ E(S).

  (ii)  (E(S), ·) is a max-min type semigroup.

  (iii)  S is inverse-free.

  (iv) G_1(S)={e}, i.e., S is action-free.

  (v)For n ∈ E_0(S) and m ∈S  (= S \ E(S)) we have n ≠ nm.



We prove the following implications: (i) ⇒ (iv) ⇒ (iii) ⇒ (v) ⇒ (i) and (iii) ⇔ (ii).


  (i) ⇒(iv). Assume Q_n ⊂ E(S) for n ∈ E_0(S). Consider any such n. Let, if possible, G_1(S) ≠{e} and  choose any m (≠ e) ∈ G_1(S) so that mn=n ≠ m. As (S,*) is a semiconvo, the associativity of `*' demands that (δ_m*δ_n)*δ_n=δ_m*(δ_n*δ_n). L.H.S. is equal to δ_mn*δ_n = δ_n*δ_n=q_n and R.H.S. is equal to δ_m*q_n. Since both sides are equal, we get q_n= q_n*δ_m. Now, m ∈(q_n*δ_m). So m ∈ Q_n, which is a contradiction. Hence, G_1(S)={e}, i.e., S is action-free.  

  (iv) ⇒ (iii). Assume that S is not inverse-free. Then G(S) does not act as the identity on E_0(S). So there exists k ∈ E_0(S) such that km ≠ k for some m (≠ e) ∈ G(S). Note that k ≠ e and also δ_k ≠ q_k since e ∈(q_k).
Now, there exists n (≠ e)  ∈ G(S)  such that mn=e. Note that kn ≠ n because if kn=n then k=e by multiplying by m on both sides, which is not true. Similarly, km ≠ m.
Consider the three different cases:
 (A) km ≠ kn,   (B) km =kn ∉ E_0(S), and   (C) km=kn ∈ E_0(S). 

 Now, because (S,*) is a semiconvo, the associativity of `*' demands that

    (δ_km*δ_kn)*δ_k = δ_km*(δ_kn*δ_k).
 


Case (A). When km ≠ kn. First, note that kn ≠ k because if kn=k then by multiplying both sides by m, we get k=km so that kn=k=km, which can not happen. The L.H.S. of (<ref>) is equal to δ_k^2mn* δ_k= q_k because k ∈ E_0(S) and R.H.S. of (<ref>) = δ_km*δ_k^2n= δ_km*δ_kn = δ_k^2mn= δ_k because km ≠ kn and k ∈ E_0(S). Since δ_k ≠ q_k, (<ref>) does not hold.  

Case (B). When km =kn ∉ E_0(S). Then L.H.S. of (<ref>) is equal to δ_k^2mn*δ_k = δ_k*δ_k=q_k because k ∈ E_0(S). And R.H.S. of (<ref>) is equal to  δ_km*δ_kn = δ_k^2mn=δ_k because kn ∉ E_0(S) and k ∈ E_0(S).  Therefore, (<ref>) does not hold.

Case (C). When km = kn ∈ E_0(S). As (S,*) is a semiconvo, the associativity of `*' demands that

    (δ_k*δ_kn)*δ_n = δ_k*(δ_km*δ_n).

Now, L.H.S.= δ_k^2n*δ_n= δ_kn*δ_n= δ_kn^2 and R.H.S.= δ_k*δ_kmn=δ_k*δ_k =q_k. Since e∈(q_k)= (R.H.S.)=(L.H.S.) = {kn^2}, we get kn^2=e, so that k ∈ G(S) and k ∈ G(S) ∩ E(S)= {e}, which is a contradiction. Therefore, (<ref>) does not hold.  
 
 Therefore, in all three cases the associativity condition  in Definition <ref> does not hold so that (S,*) is not a semiconvo, which is a contradiction to the assumption. Hence, S is inverse-free.
 
  (iii) ⇒ (v). Let, if possible, n=nm. The associativity of `*' demands that for m,n ∈ S,  (δ_n*δ_n)*δ_m = δ_n*(δ_n*δ_m). Now, as n ∈ E_0(S) and m ∈S, we get q_n*δ_m= δ_n*δ_nm= q_n.  Therefore, ∑_j ∈ Q_n q_n(j) (δ_j*δ_m)= ∑_k ∈ Q_n q_n(k) δ_k. But as m ∈S, we get 
    q_n(e) δ_m + ∑_e ≠ j ∈ Q_n q_n(j) δ_jm= q_n(e) δ_e+ ∑ _e ≠ k ∈ Q_n q_n(k) δ_k.

This shows that there exists a j_e (≠ e) ∈ Q_n such that j_em=e which implies that j_e=m=e because S is inverse-free. This is a contradiction.
 Therefore, n ≠ nm. 
 
  (v)⇒ (i). Let, if possible, there exist m ∉ E(S), which is in Q_n. Since n ≠ m, by the associativity of `*', we have  q_n*δ_m = δ_n*δ_nm. But n ≠ nm, so we have that δ_n*δ_nm=δ_n^2m= δ_nm. Therefore,  
∑_j ∈ Q_n q_n(j)(δ_j*δ_m)= δ_nm.  By the definition of `*', we get

    q_n(e) δ_m + ∑ _e ≠ j ≠ m,  j ∈ Q_n  q_n(j) δ_jm+ q_n(m) δ_m^2= δ_nm.
 So m=nm and m^2=nm. This shows that m=m^2 and hence m ∈ E(S), which is a contradiction. Therefore Q_n ⊂ E(S).

  (iii) ⇔ (ii).  Suppose (iii) holds. We know that (E(S), ·) is a subsemigroup of (S,·). Let, if possible, there exist m,n ∈ E(S) with m ≠ mn ≠ n. Then e ≠ m ≠ n ≠ e. So, by Proposition <ref>(iv), mn ∈ E_0(S). Now, the associativity of `*' demands  
    (δ_m*δ_n)*δ_mn= δ_m*(δ_n*δ_mn).
 
We see that L.H.S. = δ_mn*δ_mn= q_mn. Further, R.H.S. = δ_m*δ_nmn= δ_m*δ_mn using the fact that n ∈ E(S). As m ≠ mn we get R.H.S.= δ_m^2n, which in turn = δ_mn because m ∈ E(S). Thus q_mn= δ_mn. Now, e ∈ Q_mn so mn=e; this is not true as it shows that m,n ∈ G(S), which is a contradiction. Hence E(S) is a max-min type semigroup, i.e., (ii) holds.

The reverse implication follows from Proposition <ref>(ii). 

  Let (S,·) be a commutative discrete semigroup with identity e. Suppose (S,*) is a commutative discrete semiconvo, where `*' and other related notation and concepts are as above. Then we have the following:


  (i) For m,n ∈ E_0(S) with m ≠ n ≠ nm, we have m ∉ Q_n and Q_n · m= {nm}.

  (ii)  Under any (hence all) conditions (i)-(v) of Theorem <ref>, we have the following.


  (a) (E(S),*) is a hermitian (hence commutative) discrete hypergroup.

  (b) E(S) is finite or E(S) is isomorphic to (ℤ_+,<,max), where the order on E(S) is defined by m<n if mn=n ≠ m.

  (c) If #E(S) > 2, then for e ≠ m <n in E(S), we have the following.
 
 
  (α) q_n(e)=q_n(m) q_m(e) and 
 
  (β) q_n(e) ( 1+ ∑_e ≠ k ∈ℒ_n1/q_k(e)) ≤ 1,
 
 where for n ∈ E(S), ℒ_n := {j ∈ E(S): j<n} .

  (d) S is an ideal of (S, ·). 

  (e) For n ∈ E_0(S), m ∈S we get Q_n· m={nm}. 
 



(i). Let m,n ∈ E_0(S) with m ≠ n ≠ nm. The associativity of `*' demands that (δ_n*δ_n)* δ_m = δ_n *(δ_n *δ_m). Now L.H.S. = q_n*δ_m. Further, R.H.S. = δ_n* δ_nm which in turn = δ_nm using the fact that n ≠ nm and n ∈ E_0(S). Thus q_n*δ_m= δ_nm.  Let, if possible m ∈ Q_n. Then L.H.S. = ∑_j ∈ Q_n q_n(j) (δ_j*δ_m)= q_n(e) δ_m+ q_n(m) q_m+ ∑_e ≠ j ≠ mj ∈ Q_n q_n(j) δ_jm.  Since e ∈ Q_m and e ≠ m, the measure on L.H.S. is non-Dirac but R.H.S. is a Dirac measure, which is a contradiction. Therefore, m ∉ Q_n and hence, by q_n*δ_m= δ_nm, we get Q_n · m = {nm}.

(ii)(a).  We know that (E(S), ·) is a subsemigroup of (S, ·). Under the assumption, (i) in Theorem <ref> is satisfied. Therefore, `*' induces a product on E(S), again denoted by `*', i.e., *= *|_E(S) : E(S) × E(S) → M_F,p(E(S)). Since we assume that e ∈ Q_n, using (iii) of Theorem <ref> it is easy to see that for all m,n ∈ E(S),  e ∈(δ_m*δ_n) if and only if m=n. The associativity follows as (S,*) is a discrete semiconvo. Hence, `*' induces a hermitian discrete hypergroup structure on E(S).

(b). Under the assumption that E(S) is a max-min type semigroup, by Item <ref>(ii), it can be given a linear order by m<n if mn=n ≠ m and made a "max" semigroup with identity e. Therefore, it follows from Theorem <ref> that E(S) is finite or E(S) is isomorphic to (ℤ_+,<,max).

(c). This follows by (b) above as (E(S), ·) is a “max" semigroup and (E(S),*) is a hermitian discrete hypergroup.  

(d). Let k,n ∈S; then by the associativity of `*' we get (δ_nk*δ_n)*δ_k= δ_nk*(δ_n*δ_k) which is equivalent to δ_(nk)^2= δ_nk*δ_nk as n,k ∈S. Under the assumption S is inverse-free, it follows that nk ≠ e. This forces nk ∉ E_0(S).   Therefore, S is a subsemigroup of (S, ·). Now, to show that S is an ideal it is enough to show that for k ∈ E_0(S) and n ∈S,  nk ∈S. This can be proved in a manner similar to the proof that S̃ is a subsemigroup by using k ≠ nk and the associativity condition (δ_nk*δ_k)*δ_n= δ_nk*(δ_k*δ_n).

(e).  Let n ∈ E_0(S), m ∈S. The associativity of `*' demands that (δ_n*δ_n)*δ_m = δ_n*(δ_n*δ_m). By the definition of `*', we get q_n*δ_m=δ_n*δ_nm. Since, by (d) above, n ≠ nm, we get q_n*δ_m = δ_nm, which, using (i), shows that Q_n · m = {nm}.  


Now, we state our main theorem.
   Let (S,·) be a  commutative discrete semigroup with identity e such that S is action-free. Let `*' and other related notation and concepts be as above. Then (S, *) is a commutative discrete semiconvo if and only if the following conditions hold.
 
 
  (i)E(S) is finite or E(S) is isomorphic to (ℤ_+,<,max), where the order on E(S) is defined by m<n if mn=n ≠ m.
  
  (ii) (S, ·) is an ideal of (S, ·).  
  
  (iii) Q_n ⊂ E(S) for n ∈ E_0(S).
 
  (iv) If n ∈ E_0(S) and m ∈S then Q_n· m ={nm}.  
 
  (v) For n ∈ E_0(S), we have ℒ_n ⊂ Q_n ⊂ℒ_n ∪{n},  where for n ∈ E(S), ℒ_n := {j ∈ E(S): j<n}. 
 
  (vi)If #E(S) > 2, then for e ≠ m <n in E(S), we have the following:
 
 
  (α) q_n(e)=q_n(m) q_m(e) and 
 
  (β) q_n(e) ( 1+ ∑_e ≠ k ∈ℒ_n1/q_k(e)) ≤ 1.
 

 
 Further, under these conditions, E(S) is a hermitian discrete hypergroup. Moreover, S is a hermitian discrete hypergroup if and only if S = E(S).
 
  We assume all the conditions and prove that (S,*) is a commutative discrete semiconvo. Since S has an identity, it is enough to check the associativity of `*' to prove that (S,*) is a semiconvo. We will prove that for m,n,k ∈ S 
 
    (δ_m*δ_n)*δ_k = δ_m*(δ_n*δ_k).
 

  We divide the proof of (<ref>) into various steps.


  (i) If m,n,k ∈S, then using the fact the S is an ideal (hence subsemigroup) of S we have that the product of any two elements is in S. Therefore, both sides of (<ref>) become δ_mnk. 

  (ii) If m,n ∈S and k ∈ E(S) then both sides of (<ref>) become δ_mnk using the fact 
S is an ideal of S.    

  (iii) Suppose m ∈S and n,k ∈ E(S). If n or k is e then (<ref>) is trivial. Therefore, we can assume that n,k ∈ E_0(S).


  (a) When n=k ∈ E_0(S). 
The condition (<ref>) becomes  (δ_m*δ_n)*δ_n= δ_m*(δ_n*δ_n). Now, δ_m*(δ_n*δ_n)= δ_m*q_n = ∑_j ∈ Q_n q_n(j)   (δ_m*δ_j). Because for n ∈ E_0(S), Q_n ⊂ E(S) and m · Q_n = {mn}, we get R.H.S. of (<ref>)

    = δ_m*(δ_n*δ_n)= ∑_j ∈ Q_n q_n(j) δ_mj=  ∑_j ∈ Q_n q_n(j) δ_mn = ( ∑_j ∈ Q_n q_n(j) ) δ_mn  = δ_mn.


 Using the fact S is an ideal and n^2=n, we get L.H.S. of (<ref>) =(δ_m*δ_n)*δ_n= δ_mn. Therefore, the  associativity (<ref>)  holds. 


  (b) For n ≠ k ∈ E_0(S), R.H.S. and L.H.S. of  (<ref>) both  become δ_mnk using the fact the S is an ideal in (S, ·). Therefore, (<ref>) holds.
  
 
  (iv) We now come to cases when n ∈S and m,k ∈ E(S), or k ∈S and m,n ∈ E(S). These can be dealt with by arguments similar to those in (ii) and (iii) above.
 
 
  (v) When m,n,k ∈ E(S), associativity of `*' can be proved in a manner similar to that for proof of Theorem <ref>.
  
 Hence, (S, *) is a semiconvo.
 
Converse part of the first part of the theorem follows from Theorem <ref> and  Theorem <ref>.

Since (S,*) has a semiconvo structure and (S,·) is action-free, it  follows from Theorem <ref> that (E(S),*) has a hermitian hypergroup structure.

 Moreover, if S is hermitian hypergroup then for any e ≠ m ∈ S, e ∈(δ_m* δ_m); so we get m ∈ E_0(S) or m ∈S with m^2=e. But S is an ideal and e in not in S. So the second condition is not possible, which shows that m ∈ E_0(S). Therefore, S ⊂ E(S). Hence S=E(S).     

  
 
  (i) We can formulate a suitable analogue of Corollary <ref> based on Theorem <ref> above.
  
 
  (ii) In view of Theorem <ref> and Theorem <ref> above, identifying E(S) as a finite subset of ℤ_+ or ℤ_+ itself, for n>1,  #Q_n ≥ 2. But #Q_1 can very well be 1. As already remarked, this is the case for Dunkl-Ramirez hypergroup for a= 1/2. We note that some proofs become simpler, shorter or different when we take #Q_n > 1 for all n.
 
 
  (iii) R. C. Vrem <cit.> defined the join of two hypergroups. Because S= E(S) ⊔S, at the first  glance, our semiconvo (S,*) looks like the join of a hermitian discrete hypergroup (E(S),*) and a discrete semigroup (S, ·), but it is something different from join. 
 
  (iv) We can utilize the examples in Section <ref> to illustrate various parts of  theorems above. In particular, it is a consequence of Theorem <ref> that for Example <ref>, `*' does not induce a semiconvo structure on S. 
 
 
 The following semigroup (S, ·) is an example of a semigroup (S, ·) satisfying the conditions of Theorem <ref> and for which S is non-empty.
  For T= {1-1/r+1: r ∈ℤ_+ }, let S = T∪ℕ with `·' defined as follows.
 

    m· n  = max{m,n}   if   m  or  n ∈ T, 
     m+n    if   m,n ∈ℕ.

 
 Then S is an inverse-free commutative discrete semigroup with identity 0. Further, E(S)=T, which is isomorphic to (ℤ_+,<, ·). Also, S=ℕ is a prime ideal in S.
 



§ ACKNOWLEDGMENT



Vishvesh Kumar thanks the Council of Scientific and Industrial Research, India, for its senior research fellowship. He thanks his supervisors Ritumoni Sarma and N. Shravan Kumar for their support and encouragement.

A preliminary version of a part of this paper was included in the invited talk by Ajit Iqbal Singh at the conference “The Stone-ech compactification : Theory and Applications, at Centre for Mathematical Sciences, University of Cambridge, July 6-8 2016" in honour of Neil Hindman and Dona Strauss. She is grateful to the organizers H.G. Dales and Imre Leader for the kind invitation, hospitality and travel support. She thanks them, Dona Strauss and Neil Hindman and other participants for useful discussion. She expresses her thanks to the Indian National Science Academy for the position of INSA Emeritus Scientist and travel support. 







 
 
 
 
  
 
   
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