Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid
Felix Gotti

	
	
	
	



	
	
	Department of Mathematics
MIT
Cambridge, MA 02139
	fgotti@mit.edu
	
	
	
	
	
		If  is an ordered field and M is a finite-rank torsion-free monoid, then one can embed M into a finite-dimensional vector space over  via the inclusion M ↪(M) ↪⊗_(M), where (M) is the Grothendieck group of M. Let 𝒞 be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid M in 𝒞 are connected to the combinatorics and geometry of its conic hull (M) ⊆⊗_(M). First, we show that the submonoids of M determined by the faces of (M) account for all divisor-closed submonoids of M. Then we appeal to the geometry of (M) to characterize whether M is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in 𝒞. Along the way, we determine the cones that can be realized by monoids in 𝒞 and by finitary monoids in 𝒞.
	

Geometric and combinatorial aspects of submonoids 
 of a finite-rank free commutative monoid
    Felix Gotti
    December 30, 2023
============================================================================================















§ INTRODUCTION



Let 𝒞 denote the class containing, up to isomorphism, each monoid that can be embedded into a finite-rank free commutative monoid. If  is an ordered field and M is a monoid in 𝒞, then the chain of natural inclusions

    M ↪(M) ↪ V := ⊗_(M)

yields an embedding of M into the finite-dimensional -vector space V, where (M) is the Grothendieck group of M. Many connections between the algebraic properties of a finitely generated monoid M in 𝒞 and the geometric properties of its cone _V(M) in V have been established in the past; see, for instance, <cit.>. However, it seems that the search for similar connections for the non-finitely generated monoids in 𝒞 has only received isolated attention. In this paper we bring to the reader an initial systematic study of the connection between both arithmetic and atomic aspects of monoids M in 𝒞 and both the geometry of _V(M) and the combinatorics of the face lattice of _V(M). Here we primarily focus on the non-finitely generated members of 𝒞.

As we proceed to illustrate, various subclasses of 𝒞 consisting of non-finitely generated monoids have appeared in the literature of commutative algebra and algebraic geometry. For example, semigroups of values of singular curves were first studied by Delgado in <cit.>, and then generalized by Barucci, D'Anna, and Fröberg in <cit.> under the term “good semigroups". Good semigroups are monoids in 𝒞 that are not, in general, finitely generated (see Example <ref>). On the other hand, for a cofinite submonoid Γ of (,+) and s ∈Γ, let M^s_Γ be the monoid consisting of all arithmetic progressions of step size s that are contained in Γ. Monoids parameterized by the set of pairs (Γ, s) are members of 𝒞 known as Leamer monoids. The atomic structure of Leamer monoids is connected to the Huneke-Wiegand Conjecture (see Example <ref> for more details). Finally, let α and β be two positive irrational numbers such that α < β, and consider the submonoid M_α, β of (^2,+) that results from intersecting ^2 and the cone in the first quadrant of ^2 determined by the lines y = α x and y = β x. The monoids M_α, β, which clearly belong to 𝒞, show up in the study of Betti tables of short Gorenstein algebras (see Example <ref>).

It follows immediately that the conic hulls of the monoids M^s_Γ and M_α, β are not polyhedral cones in ^2 because they are not closed with respect to the Euclidean topology. Then Farkas-Minkowski-Weyl Theorem ensures that the monoids M^s_Γ and M_α, β cannot be finitely generated. Just as we did right now, we shall be using Farkas-Minkowski-Weyl Theorem systematically throughout this paper since it is the crucial tool guaranteeing that most of the monoids in 𝒞 we will be focused on cannot be finitely generated.

A cancellative commutative monoid is called atomic if each non-invertible element factors into irreducibles. Note that all monoids in 𝒞 are atomic. As for integral domains, a monoid is called a UFM (or a unique factorization monoid) provided that each non-invertible element has an essentially unique factorization into irreducibles. Any UFM is clearly atomic. A huge variety of atomic conditions between being atomic and being a UFM have been considered in the literature, including half-factoriality, other-half-factoriality, being finitary, and being strongly primary. In this paper, we investigate all the mentioned intermediate atomic conditions for members of 𝒞 in terms of the cones they generate.

Let M be an atomic monoid. Then M is called an HFM (or a half-factorial monoid) if for each non-invertible x ∈ M, any two factorizations of x have the same number of irreducibles (counting repetitions). On the other hand, M is called an OHFM (or an other-half-factorial monoid) if for each non-invertible x ∈ M no two essentially distinct factorizations of x contain the same number of irreducibles (counting repetitions). Also, M is called primary if it is nontrivial and for all nonzero non-invertible x,y ∈ M there exists n ∈ such that ny ∈ x + M. If M belongs to 𝒞, then M is called finitary if there exist a finite subset S of M and a positive integer n such that n (M ∖{0}) ⊆ S + M.

The rest of this paper is structured as follows. In Section <ref> we review the main concepts in commutative monoids and convex geometry we will need in later sections. In Section <ref> we present some preliminary results about monoids in 𝒞 and their cones. The main result in this section is Theorem <ref>, where we determine the cones generated by monoids in 𝒞. In Section <ref> we study the submonoids of a monoid M in 𝒞 determined by the faces of _V(M), which we call face submonoids. The most important result we achieve in this section is Theorem <ref>, where face submonoids are characterized; such a characterization allows us to deduce a characterization of the cones of primary monoids in 𝒞, which was previously obtained in <cit.>. Then in Section <ref> we offer geometric characterizations for the UFMs, HFMs, and OHFMs of 𝒞; we do this in Theorems <ref>, <ref>, and <ref>, respectively. Lastly, Section <ref> is devoted to study the cones of primary monoids and finitary monoids. The main results in this section are Theorem <ref>, which states that if a monoid M in 𝒞 is finitely primary, then the closure of the cone it generates (when =) is a rational simplicial cone, and Theorem <ref>, which states that a monoid M in 𝒞 is finitary provided that the cone it generates is polyhedral.











§ ATOMIC MONOIDS AND CONVEX CONES



In this section we introduce most of the relevant concepts related to commutative semigroups and convex geometry required to follow the results presented later. General references for any undefined terminology or notation can be found in <cit.> for commutative semigroups, in <cit.> for atomic monoids, and in <cit.> for convex geometry.








 §.§ General Notation


Set := {0,1,2,…}. If x,y ∈, then we let x,y denote the interval of integers between x and y, i.e., 

    x,y  := {z ∈| x ≤ z ≤ y}.

Clearly, x,y is empty when x > y. In addition, for X ⊆ and r ∈, we set

    X_≥ r := {x ∈ X | x ≥ r},

and we use the notation X_> r in a similar way. Lastly, if Y ⊆^d for some d ∈∖{0}, then we set Y^∙ := Y ∖{0}.









 §.§ Atomic Monoids


A monoid is commonly defined in the literature as a semigroup along with an identity element. However, we shall tacitly assume that all monoids here are also commutative and cancellative, omitting these two attributes accordingly. As we only consider commutative monoids, unless otherwise specified we will use additive notation. In particular, the identity element of a monoid M will be denoted by 0, and we let M^∙ denote the set M ∖{0}. A monoid is called reduced if its only invertible element is the identity element. Unless we state otherwise, monoids here are also assumed to be reduced.

Let M be a monoid. We write M = ⟨ S ⟩ when M is generated by a set S, i.e., S ⊆ M and no proper submonoid of M contains S. If M can be generated by a finite set, then M is said to be finitely generated. An element a ∈ M^∙ is called an atom if for each pair of elements x,y ∈ M the equation a = x+y implies that either x = 0 or y = 0. The set consisting of all atoms of M is denoted by 𝒜(M). It immediately follows that

    𝒜(M) := M^∙∖( M^∙ + M^∙).

Since M is reduced, 𝒜(M) will be contained in each generating set of M. However, 𝒜(M) may not generate M: for instance, taking M to be the submonoid of (_≥ 0,+) generated by {1/2^n | n ∈} one can see that 𝒜(M) = ∅ and, therefore, 𝒜(M) does not generate M. If 𝒜(M) generates M, then M is said to be atomic. From now on, all monoids addressed in this paper are assumed to be atomic. For x,y ∈ M, we say that x divides y in M and write x |_M y provided that y = x + x' for some x' ∈ M. An element p ∈ M^∙ is said to be prime if whenever p |_M x + y for some x,y ∈ M either p |_M x or p |_M y. The monoid M is called a UFM (or a unique factorization monoid) if each nonzero element can be written as a sum of primes (in which case such a sum is unique up to permutation). Clearly, every prime element of M is an atom. Thus, if M is a UFM, then it is, in particular, an atomic monoid.

A subset I of M is called an ideal if I + M = I (or, equivalently, I + M ⊆ I). An ideal I is principal if I = x + M for some x ∈ M. Furthermore, M satisfies the ACCP (or the ascending chain condition on principal ideals) provided that each increasing sequence of principal ideals of M eventually stabilizes. It is well known that each monoid satisfying the ACCP is atomic <cit.>. Gram's monoid, introduced in <cit.>, is an atomic monoid that does not satisfy the ACCP.

For the monoid M there exist an abelian group gp(M) and a monoid homomorphism ι M ↪gp(M) such that any monoid homomorphism ϕ M → G, where G is an abelian group, uniquely factors through ι. The group gp(M), which is unique up to isomorphism, is called the Grothendieck group[The Grothendieck group of a monoid is often called the difference or the quotient group depending on whether the monoid is written additively or multiplicatively.] of M. The monoid M is torsion-free if for all n ∈^∙ and x,y ∈ M the equality nx = ny implies that x = y. A monoid is torsion-free if and only if its Grothendieck group is torsion-free (see <cit.>). If M is torsion-free, then the rank of M, denoted by (M), is the rank of the -module gp(M), that is, the dimension of the -space ⊗_gp(M).

A multiplicative monoid F is said to be free on a subset A of F provided that each element x ∈ F can be written uniquely in the form

    x = ∏_a ∈ A a^𝗏_a(x),

where 𝗏_a(x) ∈ and 𝗏_a(x) > 0 only for finitely many a ∈ A. It is well known that for each set A, there exists a unique (up to isomorphism) monoid F such that F is a free (commutative) monoid on A. For the monoid M, we let 𝖹(M) denoted the free (commutative) monoid on 𝒜(M), and we call the elements of 𝖹(M) factorizations. If z = a_1 ⋯ a_n is a factorization in 𝖹(M) for some n ∈ and a_1, …, a_n ∈𝒜(M), then n is called the length of z and is denoted by |z|. There is a unique monoid homomorphism π𝖹(M) → M satisfying that π(a) = a for all a ∈𝒜(M). 
For each x ∈ M the set

    𝖹(x) := 𝖹_M(x) := π^-1(x) ⊆𝖹(M)

is called the set of factorizations of x. In addition, for k ∈ we set

    𝖹_k(x) := {z ∈𝖹(x) : |z| = k }⊆𝖹(M).

Observe that M is atomic if and only if 𝖹(x) is nonempty for all x ∈ M (notice that 𝖹(0) = {∅}). By <cit.>, M is a UFM if and only if |𝖹(x)| = 1 for all x ∈ M. The monoid M is called an FFM (or a finite factorization monoid) if |𝖹(x)| < ∞ for all x ∈ M. For each x ∈ M, the set of lengths of x is defined by

    𝖫(x) := 𝖫_M(x) := {|z| :  z ∈𝖹(x)}.

The sets of lengths of submonoids of (^d,+), where d ∈, have been considered in <cit.>. If |𝖫(x)| < ∞ for all x ∈ M, then M is called a BFM (or a bounded factorization monoid). Clearly, each FFM is a BFM. In addition, each BFM satisfies the ACCP (see <cit.>).

The monoid M is called finitary if it is a BFM and there exist a finite subset S of M and n ∈^∙ such that n M^∙⊆ S + M. Clearly, finitely generated monoids are finitary. On the other hand, M is called primary provided that M is nontrivial and for all x,y ∈ M^∙ there exists n ∈ such that ny ∈ x + M. If M is both finitary and primary, then it is called strongly primary.
 
A numerical monoid is a cofinite submonoid of (,+). Each numerical monoid is finitely generated and, therefore, atomic. In addition, it is not hard to see that numerical monoids are strongly primary. An introduction to numerical monoids can be found in <cit.>. The class of finitely generated submonoids of (^d,+) naturally generalizes that one of numerical monoids. Although members of the former class are obviously finitary, they need not be strongly primary; indeed, numerical monoids are the only primary monoids in this class (see Proposition <ref>). However, we shall see later that there are many non-finitely generated submonoids of (^d,+) that are primary. In addition, we will provide necessary and sufficient conditions for a submonoid of (^d,+) to be finitary.









 §.§ Convex Cones


For the rest of this section, fix d ∈^∙. As usual, for x ∈^d we let x denote the Euclidean norm of x. We always consider the space ^d endowed with the topology induced by the Euclidean norm. In addition, if  is an ordered field such that ⊆⊆, we let the vector space ^d inherit the topology of ^d. For a subset S of ^d, we let S, S̅, and S denote the interior, closure, and boundary of S, respectively.

Let  be an ordered field, and let V be a d-dimensional vector space over  for some d ∈^∙. When V = ^d, we denote the standard inner product of ^d by ⟨  , ⟩, that is, ⟨ x, y ⟩ = ∑_i=1^d x_i y_i for all x = (x_1, …, x_d) and y = (y_1, …, y_d) in ^d. The convex hull of X ⊆ V (i.e., the intersection of all convex subsets of V containing X) is denoted by _V(X). A nonempty convex subset C of V is called a cone if C is closed under linear combinations with nonnegative coefficients. A cone C is called pointed if C ∩ -C = {0}. Unless otherwise stated, we assume that the cones we consider here are pointed. If X is a nonempty subset of V, then the conic hull of X, denoted by _V(X), is defined as follows:

    _V(X) := {∑_i=1^n c_i x_i  | n ∈,  and x_i ∈ X  and c_i ≥ 0  for every i ∈ 1,n },

i.e., _V(X) is the smallest cone in V containing X. When there is no risk of ambiguity, we write (X) instead of _V(X). A cone in V is called simplicial if it is the conic hull of a linearly independent set of vectors.

Let C be a cone in V. A face of C is a cone F contained in C and satisfying the following condition: for all x,y ∈ C the fact that F intersects the open line segment {tx + (1-t)y | t ∈ and  0 < t < 1} implies that both x and y belong to F. If F is a face of C and F' is a face of F, then it is clear that F' must be a face of C. For a nonzero vector u ∈ V, consider the hyperplane H := {x ∈ V |⟨ x, u ⟩ = 0 }, and denote the closed half-spaces {x ∈ V |⟨ x, u ⟩≤ 0 } and {x ∈ V |⟨ x, u ⟩≥ 0 } by H^- and H^+, respectively. If a cone C satisfies that C ⊆ H^- (resp., C ⊆ H^+), then H is called a supporting hyperplane of C and H^- (resp., H^+) is called a supporting half-space of C. A face F of C is called exposed if there exists a supporting hyperplane H of C such that F = C ∩ H.

The cone C is called polyhedral provided that it can be expressed as the intersection of finitely many closed half-spaces. Farkas-Minkowski-Weyl Theorem states that a convex cone is polyhedral if and only if it is the conic hull of a finite set <cit.>. It is clear that every simplicial cone is polyhedral. When V = ^d, a cone in V is called rational if it is the conic hull of vectors with integer coordinates. Gordan's Lemma states that if C is a rational polyhedral cone in ^d and G is a subgroup of (^d,+), then C ∩ G is finitely generated <cit.>.

A subset A of V is called an affine set (or an affine subspace) provided that for all x,y ∈ A with x ≠ y, the line determined by x and y is contained in A. Affine sets are translations of subspaces, and an (n-1)-dimensional affine set is called an affine hyperplane. The affine hull of a subset S of V, denoted by _V(S), is the smallest affine set containing S. In addition, we let _V(S) denote the smallest subspace of V containing S. Clearly, _V(S) = _V(_V(S)). Let V' be also a finite-dimensional vector space over , and let C and C' be cones in V and V', respectively. Then C and C' are called affinely equivalent if there exists an invertible linear transformation from _V(C) to _V'(C') mapping C onto C'. When  is an intermediate field of the extension ⊆, the relative interior of S, denoted by (S), is the Euclidean interior of S when considered as a subset of _V(S). If C is a cone in ^d, then C is the disjoint union of all the relative interiors of its nonempty faces <cit.>.












§ MONOIDS IN 𝒞 AND THEIR CONES









 §.§ Monoids in the Class 𝒞


In this section we introduce the class of monoids we shall be concerned with throughout this paper. We also introduce the cones associated to such monoids.

 
	For a monoid M, the following statements are equivalent.
	
		
  (a) The monoid M can be embedded into a finite-rank free (commutative) monoid.
		
  (b) The monoid M has finite rank and can be embedded into a free (commutative) monoid.
		
  (c) There exists d ∈ such that M can be embedded in (^d,+) as a maximal-rank submonoid.
	



	(a) ⇒ (b): Suppose that F is a free (commutative) monoid of finite rank containing M, and assume that (M) ⊆(F). Then one can consider (M) as a -submodule of (F). Since (F) is a finite-rank -module, so is (M). Hence M has finite rank, which yields (b).
	(b) ⇒ (a): To argue this implication, let X be a set such that M is embedded into the free (commutative) monoid ⊕_x ∈ X x. After identifying M with its image, we can assume that M ⊆⊕_x ∈ X x and identify (M) with a subgroup of ⊕_x ∈ X x containing M. Since (M) < ∞, the dimension of the subspace W of V := ⊕_x ∈ X x generated by (M) is finite. Let {b_1, …, b_k} be a basis of W for some k ∈. For each i ∈ 1,k there exists a finite subset Y_i of X such that b_i ∈⊕_x ∈ Y_i x. As a result, W ⊆⊕_y ∈ Y y, where Y = ⋃_i=1^k Y_i. Then one finds that
	
    M ⊆( ⊕_y ∈ Y y ) ⋂( ⊕_x ∈ X x ) = ⊕_y ∈ Y y.

	Because |Y| < ∞, the monoid ⊕_y ∈ Y y is a free (commutative) monoid of finite rank containing M, and so (1) holds.
	(c) ⇒ (a): It follows trivially.
	 (a) ⇒ (c): To prove this last implication, let M be a monoid of rank d, and suppose that M is a submonoid of a free (commutative) monoid of rank r for some r ∈_≥ d. There is no loss of generality in assuming that M is a submonoid of (^r,+). Let V be the subspace of ^r generated by M. Since (M) = d, the subspace V has dimension d. Now consider the submonoid M' := ^r ∩ V of (^r,+). As M' is the intersection of the rational cone (^r ∩ V) and the lattice ^r ∩ V ≅ (^d,+), it follows from Gordan's Lemma that M' is finitely generated. On the other hand, M ⊆ M' ⊆ V guarantees that (M') = d. Since M' is a finitely generated submonoid of (^r,+) of rank d, it follows from <cit.> that M' is isomorphic to a submonoid of (^d,+). This, in turn, implies that M is isomorphic to a submonoid of (^d,+), which concludes our argument.


As we are interested in studying monoids satisfying the equivalent conditions of Proposition <ref>, we introduce the following notation.


Notation: Let 𝒞 denote the class consisting of all monoids (up to isomorphism) satisfying the conditions in Proposition <ref>. In addition, for every d ∈^∙, we set

    𝒞_d := {M ∈𝒞|(M) = d }.


A monoid is affine if it is isomorphic to a finitely generated submonoid of the free abelian group (^d,+) for some d ∈. The interested reader can find a self-contained treatment of affine monoids in <cit.>. Clearly, the class 𝒞 contains a large subclass of affine monoids. Computational aspects of affine monoids and arithmetic invariants of half-factorial affine monoids have been studied in <cit.> and <cit.>, respectively. Diophantine monoids form a special subclass of that one consisting of affine monoids; Diophantine monoids have been studied in <cit.>. Monoids in 𝒞 of small rank were recently investigated in <cit.>. Some other special subclasses of 𝒞 have been previously considered in the literature as they naturally arise in the study of algebraic curves, toric geometry, and homological algebra. Here we offer some examples.


	If M is a finitely primary monoid (see definition in Subsection <ref>), then M is primary and satisfies that M≅ (^d,+) <cit.>. Hence 𝒞 contains all finitely primary monoids. The class of finitely primary monoids contains no (nontrivial) finitely generated monoids.


 
	Good semigroups, which also form a subclass of 𝒞, were first studied in <cit.> under the name “semigroup of values" in the context of singular curves and then generalized in <cit.>, where the term “good semigroup" was coined. Good semigroups are submonoids of (^d,+) that naturally generalize semigroups of values of an algebraic curve in the sense that monoids on both classes satisfy certain common “good" properties. For instance, the semigroup S of values of the commutative ring
	
    R := [[x,y]] / ( (x^7 - x^6 + 4x^5y + 2x^3y^2 - y^4 )(x^3 - y^2) )

	is represented in Figure <ref>.
	
	As {(x,y) ∈ S | x < 13} is finite, the affine line x=13 of ^2 contains infinitely many atoms of S. Hence the good semigroup S is not finitely generated (for more details on this example, see <cit.>). In addition, it has been verified in <cit.> that the good semigroup with underlying set
	
    {(0,0)}∪{(x,y) ∈^2 | x ≥ 25  and y ≥ 27 }

	is not the semigroup of values of any algebraic curve. Good semigroups have received substantial attention since they were introduced; see for example <cit.>, and see <cit.> for more recent studies.


A subclass of non-finitely generated monoids in 𝒞 naturally shows in connection with the Huneke-Wiegand conjecture.

 
	From the structure theorem for modules over a PID, one obtains that if R is a 1-dimensional integrally-closed local domain and M is a finitely generated torsion-free R-module, then M is free if and only if M ⊗_R Hom(M, R) is torsion-free. In the same direction, Huneke and Wiegand claimed in <cit.> that if R is a 1-dimensional Gorenstein domain and M is a nonzero finitely generated non-projective R-module, then M ⊗_R Hom(M,R) has a nontrivial torsion submodule (this is known as the Huneke-Wiegand Conjecture). Given a numerical monoid Γ and s ∈∖Γ, consider the set
	
    M^s_Γ := {(0, 0)}∪{(x, n) |{x, x+s, x+2s, …, x + ns}⊆Γ}⊂^2,

	which parameterizes all arithmetic progressions of step size s contained in the monoid Γ. Clearly, M^s_Γ is a submonoid of (^2,+); it is called the Leamer monoid of Γ of step size s. Figure <ref> shows the Leamer monoid of the numerical monoid ⟨ 11,12,15 ⟩ of step size 2.
	
	The atomic structure of Leamer monoids is connected to the Huneke-Wiegand Conjecture via <cit.>. Leamer monoids are non-finitely generated <cit.> rank-2 monoids contained in the class 𝒞. Arithmetic properties of Leamer monoids have been studied in <cit.> and, more recently, in <cit.>.


The following example has been kindly provided by Roger Wiegand and should appear in the forthcoming manuscript <cit.>.

 
	Let α and β be two positive irrational numbers such that α < β, and consider the monoid M_α, β defined as follows:
	
    M_α,β := {(0,0)}⋃{ (m,n) ∈^2  | α < n/m < β}.

	Farkas-Minkowski-Weyl Theorem guarantees that M_α, β is not finitely generated and, therefore, |𝒜(M_α, β)| = ∞. In addition, M_α, β is a primary FFM (see Proposition <ref> and Proposition <ref>). The sequence of monoids (M_n)_n ≥ 3 obtained by setting
	
    α = 2/n + √(n^2 - 4)   and   β = n + √(n^2 - 4)/2

	shows up in the study of Betti tables of short Gorenstein algebras. In an ongoing project, Avramov, Gibbons, and Wiegand have proved that for each n ∈_≥ 3,
	
    𝒜(M_n) = {ω_n^1-a(1,b) | (a,b) ∈Γ},

	where ω_n  (p,q) ↦ (np - q,p) is an automorphism of M_n and Γ := × 1,n-2. This suggests the following question.
	
	
		For any irrational (or irrational and algebraic) numbers α and β with α < β, can we generalize Example <ref> to describe the set of atoms of M_α, β?
	





 §.§ The Cones of Monoids in 𝒞
 Let  be an ordered field. We say that a monoid M is embedded into a vector space V over  if M is a submonoid of the additive group V. As mentioned in the introduction, a monoid M in 𝒞 can be embedded into the finite-dimensional -vector space ⊗_(M) via

    M ↪(M) ↪⊗_(M),

where the flatness of  as a -module ensures the injectivity of the second homomorphism. Thus, one can always think of a monoid M in 𝒞 being embedded into ⊗_(M) and, therefore, one can refer to the conic hull of M.

 Let V be a finite-dimensional vector space over . A cone C in V is called positive with respect to a basis β of V provided that each v ∈ C can be expressed as a nonnegative linear combination of the vectors in β. When V = ^d for some d ∈^∙, we say that C is positive in V (without specifying any basis) if C is a positive cone with respect to the basis consisting of the canonical vectors e_1, …, e_d.

 
	Let V be a finite-dimensional vector space over an ordered field , and let M be a monoid in 𝒞 embedded into V. Then there exists a basis β of V such that _V(M) is positive with respect to β.



	Set d := (M). Since M is in 𝒞_d, Proposition <ref> guarantees the existence of a submonoid M' of (^d,+) and a monoid isomorphism φ M' → M. Extending φ to the Grothendieck groups of M' and M, and then tensoring with the -module , we obtain a linear transformation φ̅^d →_V(M) that extends φ. It is clear that φ̅(_^d(M')) = _V(M) ⊆_V(M). Because _^d(M') is a positive cone in ^d, it follows that _V(M) is positive in _V(M) with respect to the basis φ̅(e_1), …, φ̅(e_d). Finally, extending the basis φ̅(e_1), …, φ̅(e_d) to a basis β of V one obtains that _V(M) is positive in V with respect to β.


The cones of monoids in 𝒞 are pointed, as we argue now.

 
	Let V be a finite-dimensional vector space over an ordered field , and let M be a monoid in 𝒞 embedded into V. The following statements hold.
	
		
  * The cone _V(M) is pointed.
		
  * If  is an intermediate field of the extension ⊆, then the cone _V(M) is pointed.
	 



	To argue (1) suppose, without loss of generality, that V = d. By Lemma <ref> there exists a basis β of V such that _V(M) ⊆_V(β). Clearly, _V(β) is a simplicial cone and, therefore, it does not contain any one-dimensional subspace of V. This implies that _V(M) does not contain any one-dimensional subspace of V, and so _V(M) is pointed.
	To argue (2) we first notice that (with notation as in the previous paragraph) _V(β) is a closed cone of V containing _V(M). So _V(M)⊆_V(β). As observed in the previous paragraph, the fact that _V(β) is pointed forces _V(M) to be pointed.


A lattice is a partially ordered set L, in which every two elements have a unique join (i.e., least upper bound) and a unique meet (i.e., greatest lower bound). The lattice L is complete if each S ⊆ L has both a join and a meet. Two complete lattices are isomorphic if there exists a bijection between them that preserves joins and meets. For background information on (complete) lattices and lattice homomorphisms, see <cit.>.

Let V be a finite-dimensional vector space over an ordered field . For a cone C in V, the set of faces of C, denoted by 𝖥(C), is a complete lattice (under inclusion) <cit.>, where the meet is given by intersection and the join of a given set of faces is the smallest face in 𝖥(C) containing all of them. The lattice 𝖥(C) is called the face lattice of C. Two cones C and C' are combinatorially equivalent provided that their face lattices are isomorphic.

It turns out that the combinatorial and geometric structures of _V(M) do not depend on the embedding proposed in (<ref>).

 
	Let V and V' be finite-dimensional vector spaces over an ordered field , and let M and M' be isomorphic monoids in 𝒞 embedded into V and V', respectively. The following statements hold.
	
		
  * The cone _V(M) is affinely equivalent to the cone _V'(M').
		
  * The cone _V(M) is combinatorially equivalent to the cone _V'(M').
		
  * The cone _V(M) is homeomorphic to the cone _V'(M') when ⊆⊆.
	



	For cones, being affinely equivalent, being combinatorially equivalent, and being homeomorphic are all equivalence relations. Hence it suffices to argue the proposition assuming that M is a submonoid of (^d,+) and that V = ^d for some d ∈^∙.
	
	Since M ⊆^d, we can further assume that (M) ⊆^d. Let φ M → M' be a monoid isomorphism. Extending φ to the Grothendieck groups of M and M', and then tensoring with the -module , one obtains a linear transformation φ̅ V →⊗_(M') ⊆ V' that extends φ. Since  is a flat -module, φ̅ is trivial and, therefore, φ̅ is an injective linear transformation satisfying that
	
    φ̅(_V(M)) = _φ̅(V)(M') = _V'(M').

	As a consequence, the cones _V(M) and _V'(M') are affinely equivalent, from which (1) follows.
	To argue (2) it suffices to observe that because the map φ̅ is a linear bijection taking _V(M) onto _V'(M'), the assignment F ↦φ̅(F) for any face F of 𝖥(_V(M)) is an order-preserving bijection from the face lattice 𝖥(_V(M)) to the face lattice 𝖥(_V'(M')).
	To verify (3), suppose that ⊆⊆. Then part (1) guarantees the existence of an invertible linear transformation φ̅_V(_V(M)) →_V'(_V'(M')) mapping _V(M) to _V'(M'). As ⊆⊆, the subspace ^d is dense in ^d with respect to the Euclidean topology. Thus, the map φ̅ must be a homeomorphism and, therefore, _V(M) and _V'(M') are homeomorphic.


For a monoid M in 𝒞 and a finite-dimensional vector space V over an ordered field , Proposition <ref> guarantees that the geometric and combinatorial aspects, and the topological aspects (when ⊆⊆) of _V(M) do not depend on V but only on . As a consequence, we shall sometimes write _(M) instead of _V(M), choosing without specifying the finite-dimensional vector space over  where the conic hull of M is embedded into.

 
	Let  be an ordered field, and let M be a monoid in 𝒞. Then the equality _(M) = rank(M) holds.



	Consider the -vector space V := ⊗_(M). As the dimension of the -vector space ⊗_(M) equals (M), and the vector space V can be obtained from ⊗_(M) by extending scalars from  to , the equality V = (M) holds. Therefore
	
    _V(M) = _V(M) = _V((M)) = ⊗_(M) = (M).



Members of 𝒞 are finite-rank torsion-free monoids. However, not every finite-rank torsion-free monoid is in 𝒞. The next two examples shed some light upon this observation.


	A nontrivial submonoid M of (_≥ 0,+) is obviously a rank-1 torsion-free monoid. It follows from Proposition <ref> that M belongs to 𝒞 if and only if M is isomorphic to a numerical monoid. Hence <cit.> guarantees that M is in 𝒞 if and only if M is finitely generated. As a result, non-finitely generated submonoids of (_≥ 0,+) such as ⟨ 1/p | p  is prime⟩ are finite-rank torsion-free monoids that do not belong to the class 𝒞. The atomic structure of submonoids of (_≥ 0,+) have been studied in <cit.> and some of their factorization invariants have been investigated in <cit.>. In addition, their monoid algebras were recently considered in <cit.>.


Clearly, the Grothendieck group of a non-finitely generated submonoid of (_≥ 0,+) cannot be free. The following example, courtesy of Winfried Bruns, shows that a finite-rank torsion-free monoid may not belong to 𝒞 even though its Grothendieck group is free.


	Consider the additive monoid
	
    M := {(0,0)}∪{(m,n) ∈^2 | n > 0}⊆^2.

	Note that M is a submonoid of (^2,+) and, therefore, it has rank at most 2. In addition, it is clear that M is torsion-free. On the other hand,
	
    _^2(M) = {(x,y) ∈^2 | y ≥ 0},

	which is the upper closed half-space. Thus, the cone _^2(M) is not pointed. As a consequence, it follows from Proposition <ref> that M does not belong to 𝒞.









 §.§ Cones Realized by Monoids in 𝒞



Our next goal is to characterize the positive cones that can be realized as conic hulls of monoids in 𝒞. First, let us argue the following lemma.

 
	Let  be an intermediate field of the extension ⊆, and let C be a d-dimensional positive cone in ^d for some d ∈. For each x ∈  C, there exists a d-dimensional rational simplicial cone C_x such that _> 0 x ⊆  C_x and C_x ⊆{0}⋃  C.



	Take x ∈  C. For d=1, it suffices to set C_x = _≥ 0 x. Then suppose that d ≥ 2 and write x = (x_1, …, x_d) ∈^d. As C is positive and x ∈  C, one sees that x_i > 0 for every i ∈ 1,d. Let ℓ be the distance from x to the complement of C. Since C is open, ℓ > 0. Consider the d-dimensional regular simplex
	
    Δ_d := _≥ 0^d ⋂{ (y_1, …, y_d) ∈^d  | ∑_i=1^d y_i  ≤ 1 },

	and then choose N ∈ sufficiently large such that diam(Δ_d/N) = √(2)/N < ℓ. In addition, take q = (q_1,…, q_d) ∈^d_> 0 such that q_i < x_i for every i ∈ 1,d and ∑_i=1^d (x_i - q_i) < 1/N. Now set Δ := q + Δ_d/N. Clearly, x-q is an interior point of Δ_d/N and, therefore, x ∈ Δ. This, along with the fact that diam(Δ) = diam(Δ_d/N) < ℓ, ensures that Δ⊂  C. Take C_x := _^d (Δ) and observe that C_x is a polyhedral cone contained in {0}∪  C. In addition, x ∈ Δ implies that _> 0x ⊂  C_x. As Δ = d, we find that C_x = d. Hence the set of 1-dimensional faces of the polyhedral cone C_x has size at least d. On the other hand, the 1-dimensional faces of C_x are determined by some of the vertices of Δ. As the vertex q of Δ is contained in C_x, the 1-dimensional faces of C_x are precisely the d nonnegative rays containing the points q + e_i/N for each i ∈ 1,d. Thus, C_x is a d-dimensional rational simplicial cone.


Notation: Let  be an ordered field (whose prime field must be ), and take d ∈. Then we call a 1-dimensional subspace of ^d (resp., an infinite ray) a rational line (resp., a rational ray) if it contains a nonzero vector of ^d.


 
	Let  be an intermediate field of the extension ⊆, and let C be a d-dimensional positive cone in ^d for some d ∈. Then C can be generated by a submonoid of (^d,+) if and only if each 1-dimensional face of C is a rational ray.



	Set V := ^d. For the direct implication, suppose that C = _V(M), where M is a maximal rank submonoid of (^d,+). Let L be a 1-dimensional face of C, and let x be a nonzero point in L. Now take c_1, …, c_k ∈_> 0 and x_1, …, x_k ∈ M^∙ such that x = ∑_i=1^k c_i x_i. If k=1, then x_1 = 1/c_1 x ∈ L and, therefore, L is a rational ray. If k > 1, then after setting x'_1 := ∑_i=2^k c_i x_i we can see that x'_1 ∈_V(M)^∙ and
	
    c_1/1 + c_1 x_1 + 1/1 + c_1 x'_1 = 1/1 + c_1 x ∈ L.

	As L is a face of C and the open line segment from x_1 to x'_1 intersects L, the closed line segment from x_1 to x'_1 must be contained in L. In particular, x_1 ∈ L. Hence L is a rational ray.
	For the reverse implication, assume that all 1-dimensional faces of C are rational rays. Consider the set M := C ∩^d. Clearly, M is a submonoid of (^d,+) and _V(M) ⊆ C. Take x ∈ C^∙, and set ℓ := _> 0 x. As the nonempty intersection of faces of C is again a face of C, there exists a minimal (under inclusion) face C' of C containing x. Suppose, for the sake of a contradiction, that there exists a supporting hyperplane H of C' such that x ∈ H but C' ⊈H. In this case, C' ∩ H would be a face of C' containing x. However, notice that C' ∩ H is a face of C containing x that is strictly contained in C', contradicting the minimality of C'. Hence each supporting hyperplane of C' containing x also contains C'. This implies that x is in the relative interior of C'. Suppose now that C' has dimension d'. Then by Lemma <ref> there exists a rational cone C_x with C^∙_x ⊆  C' and d' 1-dimensional faces such that ℓ⊂  C_x. Now take v_1, …, v_d'∈^d ∖{0} such that the 1-dimensional faces of C_x are precisely the rays _≥ 0 v_1, …, _≥ 0 v_d'. As x ∈  C_x we can write x = ∑_i=1^d' c_i v_i for some c_1,…, c_d'∈_> 0. Because v_i ∈ C ∩^d for each i ∈ 1,d', it follows that x ∈_V(M). Hence C ⊆_V(M), which completes the proof.


The following example illustrates a positive cone in ^d that cannot be generated by any monoid in 𝒞.


	Let C be the cone in ^d generated by the set {e_1, …, e_d-1, v_d }, where v_d := π e_d + ∑_i=1^d-1 e_i. It is clear that C is a positive cone. Note, in addition, that _≥ 0 v_d is a 1-dimensional face of C. Finally, observe that _> 0 v_d contains no point with rational coordinates. Hence it follows from Theorem <ref> that C cannot be generated by any monoid in 𝒞.












§ FACES AND  DIVISOR-CLOSED SUBMONOIDS









 §.§ Face Submonoids


Let  be an ordered field, and let M be a monoid in 𝒞. We would like to understand the structure of the face lattice of _(M) in connection with the divisibility aspects of M. In particular, the submonoids of M obtained by intersecting M with the faces of _(M) are relevant in this direction as they inherit many divisibility and atomic properties from M.


	Let  be an ordered field, and let M be a nontrivial monoid in 𝒞. A submonoid N of M is called a face submonoid of M with respect to  provided that N = M ∩ F for some face F of _(M) ⊆⊗_(M).


Being a face submonoid of M with respect to  does not depend on the way M is embedded into ⊗_(M), as the following proposition indicates.

 
	Let V be a finite-dimensional vector space over an ordered field , and let M be a monoid in 𝒞 embedded into V. The following statements hold.
	
		
  * A submonoid N of M is a face submonoid with respect to  if and only if there exists a face F of _V(M) such that N = M ∩ F.
		
  * Face submonoids with respect to  are preserved by monoid isomorphisms. In particular, face submonoids do not depend on the embedding M ↪ V.
		
  * If N is a face submonoid of M with respect to , and F is a face of _V(M) satisfying that N = M ∩ F, then _V(N) = F. In particular, there exists a unique face of _V(M) determining the face submonoid N.
	



	Set V' := ⊗_(M). By the universality property of the Grothendieck group, the inclusion M ↪ V extends to an injective group homomorphism φ(M) → V and, after tensoring with the flat -module , one obtains an injective linear transformation φ̅ V' → V. For the direct implication of (1), suppose that N is a face submonoid of M, and let F' be a face of _V'(M) such that N = M ∩ F'. Clearly, F := φ̅(F') is a face of _V(M). In addition,
	
    M ∩ F = M ∩φ̅(F') = φ̅(M ∩ F') = φ̅(N) = N.

	For the reverse implication, we observe that if F is a face of _V(M) such that N = M ∩ F, then N = φ̅^-1(N) = M ∩φ̅^-1(F). As φ̅^-1(F) is a face of _V'(M), one finds that N is a face submonoid of M.
	In order to argue (2), suppose that M' is a monoid in 𝒞, and let ϕ M → M' be a monoid isomorphism. As we did in the previous paragraph, we can extend the isomorphism ϕ to a bijective linear transformation ϕ̅ W → V', where W = _V(M) and V' = ⊗_(M). Now take N to be a face submonoid of M with respect to . By the previous part, there exists a face F of _W(M) such that N = M ∩ F. Since ϕ̅(N) = ϕ̅(M) ∩ϕ̅(F) = M' ∩ϕ̅(F) and ϕ̅(F) is a face of _V'(M), it follows that ϕ̅(N) is a face submonoid of M'.
	Finally, we prove (3). Suppose that F is a face of _V(M) such that N = M ∩ F. Since N ⊆ F and F is a cone, _V(N) ⊆ F. To show the reverse inclusion, take x ∈ F^∙. Then x = ∑_i=1^k f_i x_i for x_1, …, x_k ∈ M^∙ and f_1, …, f_k ∈_> 0. If k = 1, then x_1 = x/f_1 ∈ F^∙. Otherwise, after setting x'_1 := ∑_i=2^k f_i x_i ∈_V(M) and α := 1/1 + f_1, we see that (1 - α) x_1 + α x_1' = α x ∈ F^∙. Because F is a face of _V(M) intersecting the open line segment from x_1 to x_1', both x_1 and x'_1 must belong to F^∙. Now a simple inductive argument shows that x_1, …, x_k ∈ F^∙. Therefore x_i ∈ M ∩ F = N for each i ∈ 1,k. This implies that x ∈_V(N).



The set of atoms of face submonoids can be nicely described in terms of the corresponding faces.

 
	Let V be a finite-dimensional vector space over an ordered field , and let M be a monoid in 𝒞 embedded into V. If N is a face submonoid of M determined by a face F of _V(M), then 𝒜(N) = 𝒜(M) ∩ F.



	Since 𝒜(M) ∩ F ⊆ N, one obtains that 𝒜(M) ∩ F ⊆𝒜(N). To verify the reverse inclusion, take a ∈𝒜(N) and note that a/2 ∈ F. Now take b ∈𝒜(M) such that b |_M a, and then take b' ∈ M such that a = b+b'. Because F is a face of _V(M) and a/2 belongs to the intersection of F and the open line segment {tb + (1-t)b' | t ∈ and  0 < t < 1 }, the atom b must be contained in F. As a consequence, b ∈𝒜(M) ∩ F ⊆ N. Therefore b |_M a implies that a = b ∈𝒜(M) ∩ F, which yields the desired inclusion.


For a monoid M in 𝒞, there may be submonoids of M obtained by intersecting M with certain non-supporting hyperplanes whose sets of atoms can still be obtained as in Proposition <ref>.


	Consider the submonoid M = ⟨ 2e_1, 2e_2, e_1 + e_2 ⟩ of (^2,+). It can be readily checked that 𝒜(M) = {2e_1, 2e_2, e_1 + e_2}. Now set N = M ∩ H, where H is the hyperplane (e_1 + e_2) of ^2. It is clear that N is a submonoid of M satisfying that 𝒜(N) = {e_1 + e_2} = 𝒜(M) ∩ H. However, N is not a face submonoid of M.










 §.§ Characterization of Face Submonoids


A submonoid N of a monoid M is said to be divisor-closed provided that for all y ∈ N and x ∈ M the condition x |_M y implies that x ∈ N. For any monoid M in 𝒞, the definitions of a face submonoid and a divisor-closed submonoid are equivalent.

 
	Let  be an ordered field, and let M be a monoid in 𝒞. Then a submonoid N of M is divisor-closed in M if and only if N is a face submonoid of M with respect to .



	Let k be the rank of M. Since face submonoids of M do not depend on the finite-dimensional vector space over  that M is embedded into, we can assume that M ⊆^k ⊂ V := ^k. We first verify that face submonoids of M with respect to  are divisor-closed. To do so, take a face F of _V(M) and set N := M ∩ F. To argue that N is a divisor-closed submonoid of M, take x ∈ N and y ∈ M ∖{x} with y |_M x. Then x = y + y' for some y' ∈ M, which implies that x/2 is contained in both F and the open line segment {ty +(1-t)y' | t ∈ and  0 < t < 1} from y to y'. Since F is a face of _V(M), both y and y' belong to F and, therefore, y ∈ M ∩ F = N. Hence N is divisor-closed.
	Let us argue the reverse implication by induction. Notice that when M has rank 1, it is isomorphic to a numerical monoid, and the only submonoids of M that are divisor-closed are the trivial and M itself. These are precisely the face submonoids of M in V corresponding to the origin and _V(M), respectively. Fix now k ∈_≥ 2, and assume that the divisor-closed submonoids of any monoid in 𝒞 with rank less than k are face submonoids with respect to . Let M be a maximal-rank submonoid of (^k,+), and let N be a submonoid of M that is not a face submonoid.
	
	CASE 1: (N) = k. As N is not a face submonoid of M, it follows that N ≠ M. Take v ∈ M ∖ N and a basis v_1, …, v_k ∈ N of V with v = ∑_i=1^k q_i v_i for scalars q_1, …, q_k, which are rational numbers by Cramer's rule. We assume that q_1, …, q_j ≤ 0 and q_j+1, …, q_k > 0 (not all zeros) for some index j ∈^∙. Then
	
    dv + ∑_i=1^j (dq_i) v_i = ∑_i = j+1^k (dq_i) v_i ∈ N,

	where d is the least common multiple of the denominators of all the nonzero q_i's. Since v ∉ N, the monoid N cannot be divisor-closed.
	
	CASE 2: (N) < k. Take u ∈^k such that the hyperplane
	
    H := {h ∈ V |⟨ h,u ⟩ = 0}

	of V contains linearly independent vectors v_1, …, v_k-1∈ M such that v_1, …, v_r ∈ N, where r = (N). It is clear that N ⊆ H. Now we consider the following two subcases.
	
	CASE 2.1: H is a supporting hyperplane of the cone _V(M). Consider the face F := H ∩_V(M) of _V(M). It follows from part (3) of Proposition <ref> that _V(M ∩ F) = F. Then each face of _V(M ∩ F) is a face of F and, therefore, a face of _V(M). Thus, the fact that N is not a face submonoid of M implies that N is not a face submonoid of M ∩ F. Because (M ∩ F) < (M), our inductive assumption guarantees that N is not a divisor-closed submonoid of M ∩ F. Hence N cannot be a divisor-closed submonoid of M.
	
	CASE 2.2: H is not a supporting hyperplane of the cone _V(M). In this case, there exist w_r+1, w'_r+1∈ M such that ⟨ w_r+1, u ⟩ > 0 and ⟨ w'_r+1, u ⟩ < 0. Because {v_1, …, v_k-1, w'_r+1} is a basis of V and N is a divisor-closed submonoid of M, there exists w_r+2∈ M contained in _V(v_1, …, v_k-1, w'_r+1) ∖ H and satisfying that S := {v_1, …, v_r}∪{w_r+1, w_r+2} is linearly dependent. Clearly, ⟨ w_r+2, u ⟩  < 0. After relabeling the vectors v_1, …, v_r (if necessary) and using Cramer's rule, we find that
	
    ∑_i=1^j q_i v_i = ( ∑_i=j+1^r q_i v_i ) + q_r+1 w_r+1 + q_r+2 w_r+2

	for some index j ∈ 1,r and coefficients q_1, …, q_r+1∈_≥ 0 and q_r+2∈ (not all zeros). Observe that q_r+1 and q_r+2 are both different from zero. After taking the scalar product with u in both sides of (<ref>), one can see that q_r+2⟨ w_r+2, u ⟩ = -q_r+1⟨ w_r+1, u ⟩. Hence q_r+1 and q_r+2 are both positive. Now we can multiply (<ref>) by the common denominator d of all the nonzero q_i's, to obtain that w_r+1|_M ∑_i=1^j (dq_i) v_i. Since w_r+1∉ N, the submonoid N is not divisor-closed.


We have seen in Proposition <ref> that for an ordered field  if a monoid M in 𝒞 is embedded into two -vector spaces V and V', then _V(M) is combinatorially equivalent to _V'(M). Our next goal is to argue that the combinatorial structure of _V(M) does not depend on the chosen ordered field. First, let us argue the following lemma.

 
	Let V be a finite-dimensional vector space over an ordered field , and let M be a monoid in 𝒞 maximally embedded into V. If F and F' are two faces of _V(M), then F ⊆ F' if and only if F ∩ M ⊆ F' ∩ M; in particular, F = F' if and only if F ∩ M = F' ∩ M.



	It suffices to argue the first statement, as the second statement is an immediate consequence of the first one. The direct implication is straightforward. For the reverse implication, assume that F ∩ M ⊆ F' ∩ M. Take v ∈ F. Since F ⊆_V(M), there exist f_1, …, f_n ∈_> 0 and x_1, …, x_n ∈ M^∙ such that v = ∑_i=1^n f_i x_i. If n=1, then x_1 = v/f_1 ∈ F ∩ M ⊆ F' ∩ M ⊆ F' and so x ∈ F'. Otherwise, for every j ∈ 1,n the open segment from x_j to ∑_i ≠ j f_i x_i intersects F at 1/1 + f_j v and, therefore, x_j ∈ F. As x_1, …, x_n ∈ F ∩ M ⊆ F' ∩ M ⊆ F', it follows that v ∈ F'. Hence F ⊆ F', which completes our argument.


 
	Let V and V' be finite-dimensional vector spaces over ordered fields  and 𝕂, respectively. If M and M' are isomorphic monoids in 𝒞 embedded into V and V', respectively, then _V(M) and _V'(M') are combinatorially equivalent.



	Set d := (M). By Proposition <ref> there exists a submonoid M” of (^d,+) that is isomorphic to M, i.e., M”⊆^d ⊆^d. Now set C_ := _^d(M”) and C_ := _^d(M”).
	
	We first verify that C_ and C_ are combinatorially equivalent. Clearly, F ∩^d is a face of C_ when F is a face of C_. Then define Φ𝖥(C_) →𝖥(C_) via F ↦ F ∩^d, where 𝖥(C_) and 𝖥(C_) are the face lattices of the cones C_ and C_, respectively. If F_1 and F_2 are two faces of C_ such that Φ(F_1) = Φ(F_2), then
	
    F_2 ∩ M” = Φ(F_2) ∩ M” = Φ(F_1) ∩ M” = F_1 ∩ M”.

	So it follows from Lemma <ref> that F_1 = F_2. Thus, Φ is injective. Similarly one can verify that for any two faces F_1 and F_2 of C_, the inclusion Φ(F_1) ⊆Φ(F_2) holds if and only if F_1 ⊆ F_2. As a result, Φ is a homomorphism of posets. To argue that Φ is surjective, let F' be a face of C_. As F' ∩ M” is a divisor-closed submonoid of M”, there exists a face F of C_ such that F ∩ M” = F' ∩ M”. So we obtain that Φ(F) ∩ M” = F ∩ M” = F' ∩ M”, and if follows from Lemma <ref> that Φ(F) = F'. Because Φ is a bijective homomorphism of posets, it is indeed a lattice isomorphism.
	
	Now it follows from part (2) of Proposition <ref> that _V(M) is combinatorially equivalent to C_ and, therefore, _V(M) is combinatorially equivalent to C_. In a similar manner, one can argue that _V'(M') is combinatorially equivalent to C_, whence the proposition follows.



	By virtue of Proposition <ref>, in order to study the combinatorial aspects of the cones of a monoid M in 𝒞 one can simply embed M into any finite-dimensional vector space over any ordered field. In particular, there is no loss in choosing the ordered field to be , and from now on we will do so.













§ GEOMETRY AND FACTORIALITY



In this section we study the factoriality of members of 𝒞 in connection with the geometric properties of their corresponding conic hulls. We shall provide geometric characterizations of the UFMs, HFMs, and OHFMs in 𝒞.
 

 




 §.§ Unique Factorization Monoids


To begin with, let us characterize the UFMs in 𝒞. Recall that a monoid M is a UFM if and only if |𝖹(x)| = 1 for all x ∈ M.

 
	Let M be a monoid in 𝒞, and set V = ⊗_(M). The following statements are equivalent.
	
		
  (a) The monoid M is a UFM.
		
  (b) Each face submonoid of M is a UFM.
		
  (c) The equality |𝒜(M)| =  _V(M) holds.
	



	Set d := _V(M). By part (2) of Proposition <ref>, there is not loss of generality in assuming that M is a submonoid of (^d,+), in which case V = ^d. It follows from Corollary <ref> that d = _V(M) = (M).
	(b) ⇒ (a): It is obvious.
	(a) ⇒ (c): We first verify that |𝒜(M)| ≥ d. Since _V(M) = _V(𝒜(M)), the atoms in 𝒜(M), when considered as vectors form a generating set of V. As a consequence, |𝒜(M)| ≥ V = d. To argue the reverse inequality suppose, by way of contradiction, that |𝒜(M)| > d. As the vectors in 𝒜(M) generate V, one can take distinct atoms a_1,…, a_d+1∈𝒜(M) such that a_1, …, a_d are linearly independent. Then there exist coefficients β_1, …, β_d+1∈ not all zeros such that ∑_i=1^d+1β_i a_i = 0. Since a_1, …, a_d are linearly independent and a_1, …, a_d+1∈^d, Cramer's rule guarantees that β_1, …, β_d+1∈. There is no loss in assuming that there is an index k ∈ 1,d such that β_i < 0 for each i ∈ 1,k and β_i ≥ 0 for each i ∈ k+1,d+1. Hence ∑_i=1^k β_i a_i and ∑_i = k+1^d+1 (-β_i) a_i are two distinct factorizations of the same element of M, contradicting that M is a UFM.
	(c) ⇒ (b): For this implication, suppose that |𝒜(M)| = d. Let N be a face submonoid of M. Since _V(M) = _V(𝒜(M)) and 𝒜(M) is finite, Farkas-Minkowski-Weyl Theorem ensures that _V(M) is polyhedral. Then N = M ∩ H for some supporting hyperplane H = {x ∈^d |⟨ x,u ⟩ = 0 } of _V(M) determined by u ∈^d. Suppose that _V(M) ⊆ H^-. Now if a ∈𝒜(N) and a = x_1 + x_2 for some x_1, x_2 ∈ M, then ⟨ x_1, u ⟩ + ⟨ x_2, u ⟩ = 0 and, therefore, ⟨ x_1, u ⟩ = ⟨ x_2, u ⟩ = 0. So x_1, x_2 ∈ M ∩ H = N. Because a ∈𝒜(N) either x_1 = 0 or x_2 = 0, whence a ∈𝒜(M). This implies that 𝒜(N) = 𝒜(M) ∩ N. Thus, N is a UFM.



	Let M be a monoid in 𝒞, and set V = ⊗_(M). If M is a UFM, then _V(M) is rational and polyhedral.



	As we did in the proof of Theorem <ref>, we assume that M ⊆^d and that V = ^d. By Theorem <ref>, the monoid M is finitely generated and so _V(M) is the conic hull of a finite set. Therefore it follows from Farkas-Minkowski-Weyl Theorem that _V(M) is polyhedral. In addition, _V(M) is rational because each of its 1-dimensional faces contains an element of 𝒜(M).










 §.§ Half-Factorial Monoids


The notion of half-factoriality is a weaker version of the notion of factoriality (or being a UFD). The purpose of this subsection is to offer characterizations of half-factorial monoids in the class 𝒞 in terms of their face submonoids and in terms of the convex hulls of their sets of atoms.


	An atomic monoid M is called an HFM (or a half-factorial monoid) if for all x ∈ M^∙ and z, z' ∈𝖹(x), the equality |z| = |z'| holds.


Half-factoriality was first investigated by L. Carlitz in the context of algebraic number fields; he proved that an algebraic number field is an HFD (i.e., a half-factorial domain) if and only if its class group has size at most two <cit.>. However, the term “half-factorial domain" is due to A. Zaks <cit.>. In <cit.>, Zaks studied Krull domains that are HFDs in terms of their divisor class groups. Parallel to this, L. Skula <cit.> and J. Śliwa <cit.>, motivated by some questions of W. Narkiewicz on algebraic number theory <cit.>, carried out systematic studies of HFDs. Since then HFMs and HFDs have been actively studied (see <cit.> and references therein).


HFMs in 𝒞 can be characterized as follows.

 
	Let M be a nontrivial monoid in 𝒞, and set V = ⊗_(M). The following statements are equivalent.
	
		
  (a) The monoid M is an HFM.
		
  (b) Each face submonoid of M is an HFM.
		
  (c) The inequality _V(𝒜(M)) <  _V(M) holds.
	



	By Proposition <ref>, there is not loss in assuming that M is a submonoid of (^d,+), where d := (M). In this case, V = ^d.
	(a) ⇒ (c): Suppose that M is an HFM. Since _V(𝒜(M)) = _V(M), one can take linearly independent vectors a_1, …, a_d in 𝒜(M). Take also u ∈^d and α∈ such that the polytope (a_1, …, a_d) is contained in the affine hyperplane H := {r ∈^d |⟨ r,u ⟩ = α}. In addition, fix a ∈𝒜(M), and write a = ∑_i=1^d q_i a_i for some coefficients q_1, …, q_d ∈ (by Cramer's rule, the coefficients can be taken to be rationals). The equality a = ∑_i=1^d q_i a_i, along with the fact that M is an HFM, guarantees that ∑_i=1^d q_i = 1. As a result,
	
    ⟨ a, u ⟩ = ∑_i=1^d q_i ⟨ a_i, u ⟩ = α∑_i=1^d q_i = α,

	which means that a ∈ H. Hence 𝒜(M) ⊂ H, which implies that _V(𝒜(M)) is at most d-1. Thus, _V(𝒜(M)) <  _V(M). 
	(c) ⇒ (b): Suppose that _V(𝒜(M)) <  _V(M). Then there exists an affine hyperplane H containing _V(𝒜(M)). As in the previous paragraph, take u ∈^d and α∈ such that H = {r ∈^d |⟨ r, u ⟩ = α}. Now if x ∈ M^∙ and z = a_1 + … + a_ℓ is a factorization in 𝖹(x), then
	
    ℓ = 1/α∑_i=1^ℓ⟨ a_i, u ⟩ = 1/α⟨∑_i=1^ℓ a_i, u ⟩ = 1/α⟨ x,u ⟩.

	Hence 𝖫(x) = {1/α⟨ x, u ⟩} for all x ∈ M^∙, and so M is an HFM.
	(b) ⇒ (a): It is straightforward.


 
	A nontrivial monoid M in 𝒞 is an HFM if and only if 𝒜(M) is contained in an affine hyperplane H of ⊗_gp(M) not containing the origin, in which case 𝒜(M) = M ∩ H.



	The equivalence is an immediate consequence of Theorem <ref>. It is clear that 𝒜(M) ⊆ M ∩ H. For the reverse inclusion assume, as we did in the proof of Theorem <ref>, that M ⊆^d and V = ^d (where d is the rank of M) and then take u ∈^d and α∈ such that H = { r ∈^d |⟨ r,u ⟩ = α}. As H does not contain the origin, α≠ 0. If n ∈_≥ 2 and a_1, …, a_n ∈𝒜(M), then ⟨∑_i=1^n a_i, u ⟩ = ∑_i=1^n ⟨ a_i,u ⟩ = n α≠α. Hence M ∩ H ⊆𝒜(M). 



	Fairly similar versions of Corollary <ref> have been previously established by A. Zaks in <cit.> and by F. Kainrath and G. Lettl in <cit.>.


For d ∈^∙, the convex hull in ^d of finitely many points with integer coordinates is called a lattice polytope. 


	Take d ∈^∙, and let S be a finite subset of ^d. Let P be the lattice polytope one obtains after taking the convex hull of S. Now let L be the set of all points with integer coordinates contained in P, and consider the submonoid M of (^d+1,+) defined as M = ⟨ (p,1) ∈^d+1| p ∈ L ⟩. Monoids constructed in this way are called polytopal affine monoids. Polytopal affine monoids may not be positive in general. However, they are positive when S is contained in ^d. If this is the case, then it follows from Theorem <ref> that M is an HFM in 𝒞, and it follows from Corollary <ref> that 𝒜(M) = L. Figure <ref> illustrates a lattice pentagon in the first quadrant of ^2 along with the generators in ^3 of its polytopal affine monoid.
	


The chain of implications (<ref>), where being a UFM, an HFM, and an atomic monoid are included, has received a great deal of attention since it was first studied (in the context of integral domains) by Anderson, Anderson, and Zafrullah <cit.>:

    UFM ⇒ [HFM, FFM]  ⇒ BFM ⇒ ACCP monoid ⇒ atomic monoid.

The first two implications above are obvious, while the last two implications follow from <cit.> and <cit.>. In addition, none of the implications is reversible, and examples witnessing this observation (in the context of integral domains) can be found in <cit.>. We have already seen that not every monoid in 𝒞 is an HFM. However, each monoid in 𝒞 is an FFM, as the following proposition illustrates.

 
	Each monoid in 𝒞 is an FFM.



	By Proposition <ref>, it suffices to show that for every d ∈^∙, any submonoid M of (^d,+) is an FFM. Fix x ∈ M. It is clear that ⟨ x, y ⟩≥ 0 for all y ∈ M. Thus, y |_M x implies that y≤x. As a result, the set {a ∈𝒜(M) : a |_M x} is finite, and so 𝖹(x) is also finite. Hence M is an FFM.


As an immediate consequence of Proposition <ref>, each monoid in 𝒞 satisfies the last four conditions in the chain of implications (<ref>).









 §.§ Other-Half-Factorial Monoids


Other-half-factoriality is a dual notion of half-factoriality that was introduced by Coykendall and Smith in <cit.>.


	An atomic monoid M is called an OHFM (or an other-half-factorial monoid) if for all x ∈ M^∙ and z, z' ∈𝖹(x) the equality |z| = |z'| implies that z = z'.


Although an integral domain is a UFD if and only if its multiplicative monoid is an OHFM <cit.>, in general an OHFM is not a UFM (or an HFM), as one can deduce from the next theorem.

A set of points in a d-dimensional vector space V over  is said to be affinely independent provided that for every k ∈ 2,d+1 no k of such points lie in a (k - 2)-dimensional affine subspace of V. If a set is affinely independent, its points are said to be in general linear position.

 
	Let M be a nontrivial monoid in 𝒞, and set V = ⊗_(M). The following statements are equivalent.
	
		
  (a) The monoid M is an OHFM.
		
  (b) Every face submonoid of M is an OHFM.
		
  (c) The points in 𝒜(M) are affinely independent in V.
		
  (d) The cone _V(𝒜(M)) is a simplex, whose dimension is either (M)-1 or (M).
	



	Set d := (M). By Proposition <ref>, there exists a submonoid M' of (^d,+) such that M' ≅ M. By tensoring both -modules  and (M) with , one can extend any monoid isomorphism from M' to M to a linear isomorphism from ^d to V. Since the property of being affinely independent is clearly preserved by isomorphisms, there is no loss of generality in assuming that M is a submonoid of (^d,+), and we do so.
	(a) ⇒ (c): Assume that M is an OHFM and suppose, by way of contradiction, that the points in 𝒜(M) are not affinely independent. Then there exist k ∈ 2,d+1 and distinct vectors a_1, …, a_k ∈𝒜(M) contained in a (k-2)-dimensional affine subspace W of V. Let M' denote now the submonoid of M generated by the atoms a_1, …, a_k. Since W - a_k is a (k-2)-dimensional subspace of V, the vectors a_1 - a_k, …, a_k-1 - a_k are linearly dependent in W - a_k. This, along with Cramer's rule, guarantees that  ∑_i=1^k-1 q_i (a_i - a_k) = 0 for some rational coefficients q_1, …, q_k-1 (not all zeros). After relabeling vectors and coefficients, we can assume the existence of j ∈ 1,k-2 such that q_1, …, q_j are negative and q_j+1, …, q_k-1 are nonnegative. Set
	
    x := ∑_i=1^j (-mq_i) a_k + ∑_i=j+1^k-1(mq_i)a_i ∈ M',

	where m is the least common multiple of the denominators of all the nonzero q_i's. Then
	
    z := (∑_i=1^j (-mq_i) )a_k + ∑_i=j+1^k-1(mq_i)a_i    and    z' :=   ∑_i=1^j (-mq_i)a_i + (∑_i=j+1^k-1 (mq_i) ) a_k

	are two factorizations in 𝖹_M'(x) having the same length. As a_1, …, a_k are also atoms of M, it follows that z and z' are also factorizations in 𝖹_M(x), which contradicts that M is an OHFM.
	(c) ⇒ (a): Suppose that the points in 𝒜(M) are affinely independent in V. We have seen in the proof of Theorem <ref> that |𝒜(M)| ≥ _V(M). So |𝒜(M)| ∈{d, d+1}. If |𝒜(M)| = d, then Theorem <ref> ensures that M is a UFM and, therefore, an OHFM. Thus, we assume that |𝒜(M)| = d+1. Let 𝒜(M) =: {a_0, a_1, …, a_d} and suppose, by way of contradiction, that M is not an OHFM. This guarantees the existence of two distinct nonzero (d+1)-tuples (m_0, …, m_d) and (n_0, …, n_d) in ^d satisfying that
	
    ∑_i=0^d m_i a_i = ∑_i=0^d n_i a_i    and   ∑_i=0^d (m_i - n_i) = 0.

	Assume, without loss of generality, that m_0 ≠ n_0. Let H be an affine hyperplane in V containing a_1, …, a_d. Take u ∈^d and α∈ such that H = {r ∈^d |⟨ r,u ⟩ = α}. As the points a_0, a_1, …, a_d are affinely independent, a_0 ∉ H. Then
	
    0  = ∑_i=0^d (m_i - n_i) ⟨ u, a_i ⟩ = (m_0 - n_0) ⟨ u, a_0 ⟩ + α∑_i=1^d (m_i - n_i) = (m_0 - n_0) (⟨ u, a_0 ⟩ - α).

	As a result, ⟨ u, a_0 ⟩ = α, which contradicts that a_0 does not belong to H. Hence M must be an OHFM.
	(c) ⇒ (b): Suppose that the points in 𝒜(M) are affinely independent. Let N be a face submonoid of M, and let F be a face of _V(M) satisfying that N = M ∩ F. Proposition <ref> ensures that 𝒜(N) = 𝒜(M) ∩ F. Since the set of points 𝒜(M) is affinely independent, the set of points 𝒜(N) is also affinely independent. As we have already proved the equivalence of (a) and (c), we can conclude that N is an OHFM. Hence statement (b) follows.
	(b) ⇒ (a): It is clear.
	(c) ⇔ (d): These two statements are obviously restatements of each other.


 
	Let N be a numerical monoid. Then N is an OHFM if and only if the embedding dimension of N is at most 2.



	Theorem <ref> was indeed motivated by Corollary <ref>, which was first proved by Coykendall and Smith in <cit.>.


The fact that every proper face submonoid of a monoid M in 𝒞 is an OHFM does not guarantee that M is an OHFM, as one can see in the following example.


	Consider the submonoid M := ⟨ 2e_1, 3e_1, 2e_2, 3e_2 ⟩ of (^2,+). It is easy to argue that 𝒜(M) = {2e_1, 3e_1, 2e_2, 3e_2}. Observe that the 1-dimensional faces of _^2(M) are _≥ 0 e_1 and _≥ 0 e_2. Then there are two face submonoids of M corresponding to 1-dimensional faces of _^2(M), and they are both isomorphic to the numerical monoid ⟨ 2, 3 ⟩, which is an OHFM by Corollary <ref>. Hence every proper face submonoid of M is an OHFM. However, _^2(𝒜(M)) is not a simplex and, therefore, it follows from Theorem <ref> that M is not an OHFM. 


We conclude this section with the following proposition.

 
	Let M be an OHFM in 𝒞, and set V = ⊗_(M). Then the faces of _V(M) whose corresponding face submonoids are not UFMs form a (possibly empty) interval in the face lattice 𝖥(_V(M)).



	In light of Proposition <ref> and Proposition <ref>, one can assume that M is a submonoid of (^d,+), where d := (M). In this case, V = ^d.
	
	Let 𝒩 consist of all faces of _V(M) whose corresponding face submonoids are not UFMs. If M is a UFM, it follows from Theorem <ref> that every face submonoid of M is also a UFM and, therefore, 𝒩 is empty. So we assume that M is not a UFM.
	
	Among all the faces in 𝒩, let F and F' be minimal in 𝖥(_V(M)). Suppose, by way of contradiction, that F ≠ F'. Set N := M ∩ F and N' := M ∩ F'. It follows from Proposition <ref> that F = _V(N) and F' = _V(N'). Since F and F' are minimal, they are not comparable and so we can take a ∈𝒜(N) ∖𝒜(N') and a' ∈𝒜(N') ∖𝒜(N). Once again, one can rely on the minimality of F and F' to obtain
	
    (𝒜(N)) =  (𝒜(N) ∖{a})    and    (𝒜(N')) =  (𝒜(N') ∖{a'}).

	As a result, the rank of the set 𝒜 := 𝒜(N) ∪𝒜(N') is at most |𝒜|-2. Set n := |𝒜|, and let A be the d × n real matrix whose columns are the vectors in 𝒜 (after some order is fixed). Then A = (𝒜) ≤ n-2. Thus, A ≥ 2. Consider the hyperplane of ^n defined by
	
    H := {(x_1, …, x_n) ∈^n  | ∑_i=1^n x_i = 0 }

	and notice that
	
    (H ∩ A) =  H +  A -  span(H ∪  A) ≥ 1.

	Therefore there is a nonzero vector (q_1, …, q_n) ∈ A satisfying that ∑_i=1^n q_i = 0. First, taking j ∈ 1,n such that  q_1, …, q_j ≤ 0 and q_j+1, …, q_n > 0, then taking m to be the least common multiple of the denominators of all the nonzero q_i's, and finally proceeding as we did in the second paragraph of the proof of Theorem <ref>, we can obtain two distinct factorizations of the same element of M having the same length. However, this contradicts that M is an OHFM. Hence there exists only one minimal face of _V(M) whose face submonoid is not a UFM, namely, F. Clearly, the face submonoid of any face containing F cannot be a UFM. This implies that [F, _V(M)] ⊆𝒩. The reverse inclusion follows from the uniqueness of a minimal face in 𝒩. Hence 𝒩 is the interval [F, _V(M)].


The reverse implication of Proposition <ref> does not hold, as the following example illustrates.


	Consider the submonoid M := ⟨ 3e_1, 3e_2, 2e_3, 3e_3 ⟩ of (^3, +). It can be readily verified that 𝒜(M) = { 3e_1, 3e_2, 2e_3, 3e_3 }. Since {2e_3, 3e_3} is an affinely dependent set, it follows from Theorem <ref> that M is not an OHFM. However, the non-UFM face submonoids of M are precisely those determined by the faces of _V(M) contained in the interval [ e_3, _V(M)]. The face lattice of _^3(M) is shown in Figure <ref>.
	













§ CONES OF PRIMARY MONOIDS AND FINITARY MONOIDS



 As mentioned at the beginning of this paper, the classes of primary monoids and finitary monoids have been crucial in the development of factorization theory as the arithmetic structure of their members abstracts certain properties of important classes of integral domains. The first part of this section is devoted to investigate some geometric aspects of primary monoids in 𝒞. Then we shift our focus to the study of finitary monoids of 𝒞.






 §.§ Primary Monoids


Recall that a monoid M is primary if M is nontrivial and for all x,y ∈ M^∙ there exists n ∈ such that ny ∈ x + M. The study of primary monoids was initiated by T. Tamura <cit.> and M. Petrich <cit.> in the 1970s and has received a great deal of attention since then <cit.>. Primary monoids naturally appear in commutative algebra: an integral domain is 1-dimensional and local if and only if its multiplicative monoid is primary <cit.>.

The primary monoids in 𝒞 are precisely those minimizing the number of face submonoids.

 
	Let M be a nontrivial monoid in 𝒞, and set V = ⊗_(M). The following statements are equivalent.
	
		
  (a) The monoid M is a primary monoid.
		
  (b) The only face submonoids of M are {0} and M.
		
  (c) The cone _V(M)^∙ is an open subset of V.
	



	By using Proposition <ref> and Proposition <ref>, we can assume that M is a submonoid of (^d,+), where d := (M). In this case, V = ^d.
	(a) ⇔ (b): It follows from <cit.> that M is primary if and only if the only divisor-closed submonoids of M are {0} and M. This, along with Theorem <ref>, implies that the conditions (a) and (b) are equivalent.
	(b) ⇒ (c): Take x ∈_V(M)^∙. Since _V(M) is the disjoint union of the relative interiors of all its faces, there exists a face F of _V(M) such that x ∈  F. As x ≠ 0, the dimension of F is at least 1 and, therefore, M ∩ F is a nontrivial face submonoid of M. It follows now from part (b) that M ∩ F = M. As a consequence, x belongs to the relative interior of _V(M). Hence _V(M)^∙ is open.
	(c) ⇒ (b): Since every proper face of _V(M) is contained in the boundary of _V(M), the fact that _V(M)^∙ is open implies that the only proper face of _V(M) is the origin, from which (b) follows.



	We want to emphasize that the equivalence (a) ⇔ (c) in Proposition <ref> was first established by Geroldinger, Halter-Koch, and Lettl <cit.>. However, our approach here is quite different as we have obtained the same result based primarily on the combinatorial structure of the face lattice of _V(M).


Primary monoids in 𝒞 account for all primary submonoids of any (non-necessarily finite-rank) free (commutative) monoid, as the next proposition illustrates.


	Let M be a primary submonoid of a free (commutative) monoid. Then M has finite rank, and M can be embedded into (^d,+), where d := (M).



	Let F_P be a free (commutative) monoid on an infinite set P such that M is a submonoid of F_P. For s ∈ F_P and S ⊆ F_P, write
	
    Spec(s) := {p ∈ P | p  divides  s  in  F_P }   and   Spec(S) := ⋃_s ∈ SSpec(s).

	Suppose, by way of contradiction, that Spec(M) contains infinitely many elements. Fix x ∈ M^∙. Since F_P is free and, therefore, a UFM, the set Spec(x) must be finite. Then we can take p ∈ P such that p ∈Spec(M) ∖Spec(x). Because p is a prime element of F_P, it is not hard to verify that the set S := {y ∈ M | p  does not divide  y  in  F_P } is a divisor-closed submonoid of M. The fact that p ∉Spec(x) implies that S is a nonzero submonoid of M, and the fact that p ∈Spec(M) implies that S is a proper submonoid of M. Therefore the monoid M contains a proper nonzero divisor-closed submonoid, which contradicts that M is primary. As a consequence, Spec(M) must be finite, and so M can be naturally embedded into the finite-rank free (commutative) monoid ⊕_i=1^t  p_i, where p_1, …, p_t are the prime elements in Spec(M). It follows now from Proposition <ref> that M can be embedded into (^r,+).










 §.§ Finitely Primary Monoids



Now we restrict our attention to a special subclass of primary monoids that has been key in the development of factorization theory, the class consisting of finitely primary monoids. The initial interest in this class originated from commutative algebra: the multiplicative monoid of a 1-dimensional local Mori domain with nonempty conductor is finitely primary <cit.>. Finitely primary monoids were introduced in <cit.> by Geroldinger.


The complete integral closure of a monoid M, denoted by M, is defined as follows:

    M := {x ∈gp(M) | there exists  y ∈ M  such that  nx + y ∈ M  for every  n ∈}.

Clearly, M is a submonoid of gp(M) containing M, and so (M) = (M). A monoid M is called finitely primary if there exist d ∈ and a UFM F := ⟨ p_1, …, p_d ⟩, where p_1, …, p_d are pairwise distinct prime elements in F, such that the following conditions hold:

	
  * the monoid M is a submonoid of F,
	
  * the inclusion M^∙⊆ p_1 + … + p_d + F holds, and
	
  * the inclusion α(p_1 + … + p_d) + F ⊆ M holds for some α∈^∙.

In this case, it follows from <cit.> that M≅ (^d,+). Then (M) = d and, moreover, any finitely primary monoid of rank d belongs to 𝒞_d. On the other hand, it also follows from <cit.> that finitely primary monoids are primary. Therefore Proposition <ref> guarantees that for any finitely primary monoid M the set _V(M)^∙ is open in the finite-dimensional -vector space V := ⊗_(M). This implies, in particular, that nontrivial finitely primary monoids cannot be finitely generated. As the following theorem reveals, the closures of their cones happen to be simplicial cones.

 
	Let M be a finitely primary monoid, and set V = ⊗_(M). Then M belongs to 𝒞, and the set _V(M) is a rational simplicial cone in V.



	Let d be the rank of M. We have already observed that M is in the class 𝒞. Then by virtue of Proposition <ref> and Proposition <ref>, we can assume that M is a submonoid of (^d,+) and, therefore, that V = ^d. In addition, there is no loss in assuming that M ⊆M⊆^d.
	
	Because M≅ (^d,+), one can take distinct prime elements p_1, …, p_d of M such that M = ⟨ p_1, …, p_d ⟩ = ⊕_i=1^d  p_i. It follows from <cit.> that 
	
    M^∙⊆ p_1 + … + p_d + M  and  α(p_1 + … + p_d) + M⊆ M

	for some α∈^∙. Let C_p be the cone in V generated by p_1,…, p_d. Clearly, C_p is a rational simplicial cone of dimension d. We claim that _V(M) = C_p. Since
	
    M^∙⊆ p_1 + … + p_d + M⊂  C_p,

	the inclusion M ⊆ C_p holds. As a consequence, _V(M) ⊆ C_p and, as the cone C_p is a closed set in V, the inclusion _V(M)⊆ C_p follows. Let us proceed to argue that C_p ⊆_V(M). To do so, fix ϵ > 0 and fix also an index j ∈ 1,d. Let L be the 1-dimensional face of C_p in the direction of the vector p_j, and consider the conical open ball with central axis L given by
	
    B(p_j, ϵ) := { w ∈ V ∖{0} | w - 𝗉_L(w)/w < ϵ},

	where 𝗉_L  V → p_j is the linear projection of V onto its subspace p_j. It is clear that the set {0}∪( B(p_j, ϵ) ∩  C_p ) is a d-dimensional subcone of C_p and, therefore, it must intersect M. Then one can take y ∈M∩  C_p such that _> 0 y ⊂ B(p_j, ϵ). Because
	
    α( M∩  C_p ) ⊆α(p_1 + … + p_d) + M⊆ M,

	α y ∈ M. As a result, _> 0 y ⊂_V(M). As _V(M) and every open conical ball with central axis L have an open ray in common, p_j ∈ L ⊆_V(M). Because the index j was arbitrarily taken, p_j ∈_V(M) for every j ∈ 1,d, and so C_p ⊆_V(M). Hence _V(M) is a rational simplicial cone.


With notation as in Theorem <ref>, the facts that M is primary and _V(M) is a rational simplicial cone do not guarantee that M is finitely primary. The following example sheds some light upon this observation.


	Consider the subset M of ^2 defined by
	
    M := {(0,0)}∪{ (n,m) ∈^2 | n,m ∈^∙ and  m ≤ 2^n }.

	From the fact that f(x) = 2^x is a convex function, one can readily verify that M is a submonoid of (^2,+). Since M contains (n,1) for every n ∈^∙, the ray _≥ 0 e_1 is contained in _^2(M). On the other hand, the fact that {(n, 2^n) | n ∈^∙}⊂ M, along with lim_n →∞ 2^n/n = ∞, guarantees that the ray _≥ 0 e_2 is contained in _^2(M). Thus, _^2(M) is the closure of the first quadrant and so
	
    _^2(M) = { (0,0)}∪{(x,y) ∈^2 | x,y > 0 } = {(0,0)}∪^2_> 0.

	Because _^2(M)^∙ is an open set in ^2, it follows from Proposition <ref> that M is a primary monoid. On the other hand, the equality _^2(M) = ^2_≥ 0 holds, and so _^2(M) is a rational simplicial cone.
	
	To argue that M is not finitely primary, it suffices to verify that M≇(^2,+). To do so, fix m ∈, and then take N ∈ large enough so that nm ≤ 2^n for every n ≥ N. Note that y := (N,Nm) belongs to M. Moreover,
	
    n(1,m) + y = (n+N, (n+N)m) ∈ M

	for every n ∈. Therefore (1,m) ∈M for every m ∈. On the other hand, for any m ∈^∙ and (a,b) ∈ M^∙,
	
    2^a(0,m) + (a,b) = (a, 2^am + b) ∉ M.

	Hence (n,m) ∈M^∙ implies that n > 0. As a result, M = {(n,m) ∈^2 | n > 0}. Since 𝒜(M) = {(1,n) | n ∈} contains infinitely many elements, M≇(^2,+). Consequently, M cannot be finitely primary.









 §.§ Finitary Monoids


Let M be a monoid. Recall that M is finitary if it is a BFM and there exist a finite subset S of M and a positive integer n satisfying that n M^∙⊆ S + M. Geroldinger et al. introduced and studied the class of finitary monoids in <cit.> motivated by the fact that monoids in this class naturally show in commutative ring theory: the multiplicative monoid of a Noetherian domain R is finitary if and only if R is 1-dimensional and semilocal <cit.>. In addition, finitary monoids conveniently capture certain aspects of the arithmetic structure of more sophisticated monoids, including v-Noetherian G-monoids <cit.> and congruence monoids <cit.>. Every finitely generated monoid is finitary. In particular, affine monoids are finitary.

The monoid M is said to be weakly finitary if there exist a finite subset S of M and n ∈^∙ such that nx ∈ S + M for all x ∈ M^∙. Clearly, every finitary monoid is weakly finitary. The face submonoids of a monoid in 𝒞 inherit the condition of being (weakly) finitary.

 
	Let M be a monoid in 𝒞. Then M is finitary (resp., weakly finitary) if and only if each face submonoid of M is finitary (resp., weakly finitary).



	We will prove only the finitary version of the proposition as the weakly finitary version follows similarly. By Proposition <ref> we can assume that M is a submonoid of (^d,+), where d is the rank of M.
	
	Suppose that M is finitary. Take F to be a face of _^d(M), and consider the face submonoid N := M ∩ F. Since M is finitary, there exist n ∈ and a finite subset S of M such that n M^∙⊆ S + M. We claim that n N^∙⊆ S_F + N, where S_F := S ∩ F. Take x_1, …, x_n ∈ N^∙ = M^∙∩ F. Because n (M^∙∩ F) ⊆ n M^∙⊆ S + M, there exist s ∈ S and y ∈ M such that ∑_i=1^n x_i = s + y. Since M ∩ F is a divisor-closed submonoid of M, we find that s,y ∈ F. Therefore s ∈ S_F and y ∈ N; this implies that ∑_i=1^n x_i ∈ S_F + N. Hence N is a finitary monoid. The reverse implication follows trivially as _^d(M) is a face of itself.


Our next goal is to give a sufficient geometric condition for a monoid in 𝒞 to be finitary. First, let us recall the notion of triangulation. Let V be a finite-dimensional vector space over an ordered field . A conical polyhedral complex  in V is a collection of polyhedral cones in V satisfying the following conditions:

	
  * every face of a polyhedron in  is also in , and
	
  * the intersection of any two polyhedral cones C_1 and C_2 in  is a face of both C_1 and C_2.

Clearly, the underlying set of the face lattice of a given polyhedral cone is a conical polyhedral complex. For a conical polyhedral complex  in V, we set || := ⋃_C ∈  C. Let  and ' be two conical polyhedral complexes. The complex ' is said to be a polyhedral subdivision of  provided that || = |'| and each face of  is the union of faces of '. A polyhedral subdivision 𝖳 of  is called a triangulation of  provided that 𝖳 consists of simplicial cones. Every conical polyhedral complex has the following special triangulations.

<cit.> 
	Let  be a conical polyhedral complex in ^d for some d ∈^∙, and let S ⊂ || be a finite set of nonzero vectors such that S ∩ C generates C for each C ∈. Then there exists a triangulation 𝖳 of  such that {_≥ 0v | v ∈ S} is the set  of 1-dimensional faces of 𝖳.


We are in a position now to offer a sufficient geometric condition for a monoid in 𝒞 to be finitary.

 
	Let M be a monoid in 𝒞, and set V = ⊗_(M). If _V(M) is polyhedral, then M is finitary.



	Let d be the rank of M. Based on Proposition <ref> and Proposition <ref>, one can assume that M ⊆^d. In this case, V = ^d. Since _V(M) is polyhedral, it follows by Farkas-Minkowski-Weyl Theorem that _V(M) is the conic hull of a finite set of vectors. As the vectors in such a generating set are nonnegative rational linear combinations of vectors in M, there exists S = {v_1, …, v_k}⊂ M with k ≥ d such that _V(M) = _V(S). By Theorem <ref>, there exists a triangulation 𝖳 of the face lattice of _V(M) whose set of 1-dimensional faces is {_≥ 0 v_i | i ∈ 1,k }. Then for any T ∈𝖳 there are unique indices t_1, …, t_d satisfying that
	
    1 ≤ t_1 < … < t_d ≤ k    and    T = _V(v_t_1, …, v_t_d),

	and we can use this to assign to T the parallelepiped
	
    Π_T := {∑_i=1^d α_1 v_t_i | 0 ≤α_i < 1  for every i ∈ 1,d }.

	It is clear that
	
    |Π_T ∩^d| < ∞   and   Π_T ∩^d ⊂∑_i=1^d _≥ 0 v_t_i.

	Then we can choose N_T ∈ sufficiently large so that N_T v ∈∑_i=1^d  v_t_i for every v ∈Π_T ∩^d. Now take m := max{N_T   |Π_T ∩^d| : T ∈𝖳} and set n := m   |𝒯|. In order to show that M is finitary, it suffices to verify that nM^∙⊆ S + M.
	
	Take (possibly repeated) elements x_1, …, x_n ∈ M^∙. For every x ∈{x_1,…, x_n}, there exists T ∈𝖳 with x ∈ T. Let T = _V(v_t_1, …, v_t_d) for t_1 < … < t_d be a simplicial cone in 𝖳. Observe that we can naturally partition T into (translated) copies of the parallelepiped Π_T, that is, T equals the disjoint union of the sets v + Π_T for every v ∈∑_i=1^d v_t_i. As a result, there exist z ∈Π_T ∩^d and coefficients α_1, …, α_d ∈ satisfying that
	
    x = z + ∑_i=1^d α_i v_t_i.

	Hence for i ∈ 1,n, we can write x_i = z_i + m_i for some z_i ∈⋃_T ∈𝖳Π_T ∩^d and m_i ∈ M. Since n = m |𝖳|, there exists T_0 ∈𝖳 such that
	
    |{i ∈ 1,n | z_i ∈Π_T_0∩^d}| ≥ m.

	Consider now the equivalence relation on the set of indices {i ∈ 1,n | z_i ∈ T_0} defined by i ∼ j whenever z_i = z_j. The fact that m ≥ N_T_0  |Π_T_0∩^d| guarantees the existence of a class I determined by the relation ∼ and containing at least N_T_0 distinct indices. Take I_0 ⊆ I such that |I_0| = N_T_0. Setting z := z_i for some i ∈ I_0, we see that
	
    ∑_i ∈ I_0 z_i = N_T_0z ∈ v_1 + … +  v_n ∈ S + M

	and, therefore, ∑_i ∈ I_0 z_i = v + m for some v ∈ S and m ∈ M. As a result, one can set m' := ∑_i=1^n x_i - ∑_i ∈ I_0 x_i ∈ M to obtain that
	
    ∑_i=1^n x_i = ( ∑_i ∈ I_0 x_i ) + m' = ∑_i ∈ I_0 z_i + m' + ∑_i ∈ I_0 m_i = v + (m + m' + ∑_i ∈ I_0 m_i ) ∈ S + M.

	Since the elements x_1, …, x_n were arbitrarily taken in M^∙, the inclusion n M^∙⊆ S + M holds. Hence the monoid M is finitary.


According to the characterization of cones generated by monoids in 𝒞 we have provided in Theorem <ref>, every d-dimensional positive cone C of ^d with C^∙ open can be generated by a monoid in 𝒞. Indeed, any such a cone can be generated by a finitary monoid in 𝒞.

 
	For d ∈, let C be a positive cone in ^d. If C^∙ is open in ^d, then C can be generated by a finitary monoid in 𝒞.



	Assume that C^∙ is open in ^d. Take M = ^d ∩ C. It is clear that C = _^d(M). Now take v_0 ∈ M^∙, and consider the monoid M' := {0}∪ (v_0 + M). Let C' be the cone generated by M'. Notice that ^d ∩ C and ^d ∩ C' are the cones generated by M and M' over , respectively. So proving that C' = C amounts to showing that ^d ∩ C' = ^d ∩ C (see <cit.>). Since M' ⊆ M it follows that ^d ∩ C' ⊆^d ∩ C. Now let ℓ_0 be the distance from {v_0} to ^d ∖ C. As ^d ∖ C is closed and {v_0} is compact, ℓ_0 > 0. Now take v ∈^d ∩ C^∙ such that v > 1, and let ℓ be the distance from v to ^d ∖ C. By a similar argument, ℓ > 0. Notice that the conical ball
	
    B(v, ℓ) := { w ∈^d  | w - 𝗉_v(w)/w < ℓ/2}

	is contained in ^d ∩ C; here 𝗉_v ^d → v is the projection of ^d onto its one-dimensional subspace v. Take N ∈ such that
	
    N > max{v_0/v - 1, 2 v_0/ℓ}

	and Nv ∈^d. Now set w_0 := Nv - v_0. Notice that w_0≥ N v - v_0 > N. Then
	
    w_0 - 𝗉_v(w_0)/w_0 < v_0 - 𝗉_v(v_0) /N≤v_0/N < ℓ/2.

	Hence w_0 ∈^d ∩ B(v,ℓ) ⊆^d ∩ C, and so there exist coefficients c_1, …, c_k ∈_> 0 and elements v_1, …, v_k ∈ M^∙ such that w_0 = ∑_i=1^k c_i v_i. As a consequence, one has that nv ∈ M' ⊆_V(M') for some n ∈ and, therefore, v ∈^d ∩_V(M'). Hence ^d ∩ C^∙⊆^d ∩ C.
	
	As M' generates C, we only need to verify that M' is finitary. Take w_1, w_2 ∈ M'^∙, and then v_1, v_2 ∈ M such that w_1 = v_0 + v_1 and w_2 = v_0 + v_2. Then
	
    w_1 + w_2 = v_0 + (v_0 + v_1 + v_2) ∈ v_0 + (v_0 + M) ⊂ v_0 + M'.

	As a result, 2M'^∙⊆ v_0 + M', which implies that M' is a finitary monoid.


Theorem <ref> and Proposition <ref> indicate that there is a huge variety of finitary monoids in 𝒞. We proceed to exhibit a monoid in 𝒞_2 that is not even weakly finitary. First, let us introduce the following notation.


Notation: For x ∈^2_≥ 0∖{(0,0)}, we let (x) ∈_≥ 0∪{∞} denote the slope of the line x, and for X ⊂_≥ 0^2 we set

    (X) := {(x) | x ∈ X^∙}.



	Construct a sequence (v_n)_n ≥ 1 of vectors in ^∙×^∙ as follows. Set v_1 = (1,1) and suppose that for some n ∈  and for every i ∈ 1,n-1 we have chosen vectors v_i = (x_i, y_i) ∈^∙×^∙ satisfying that (v_i) < (v_i+1) and i v_i < v_i+1. Then take v_n+1 = (x_n+1, y_n+1) ∈^2 in such a way that x_n+1 > 0, (v_n+1) > (v_n), and v_n+1 > n v_n. Now one can consider the additive submonoid M := ⟨ v_n | n ∈^∙⟩ of ^2. Clearly, 𝒜(M) ⊆{v_n | n ∈^∙}. On the other hand, the fact that v_m > v_n when m > n implies that only atoms in {v_1, …, v_n-1} can divide v_n in M. This, along with the fact that
	
    (v_n) > max{(v_i) | i ∈ 1,n-1 }

	for every n ∈, guarantees that 𝒜(M) = {v_n | n ∈^∙}. Finally, let us verify that M is not weakly finitary. Assume for a contradiction that there exist n ∈ and a finite subset S of M such that nx ∈ S + M for all x ∈ M^∙. We can assume without loss of generality that S ⊆𝒜(M), so we let S = {v_n_1, …, v_n_k}, where n_1 < … < n_k. Take N > max{n, n_k}. Then write n' v_N = v_n_i + m for some i ∈ 1,k and m ∈ M such that n' ≤ n and v_N ∤_M m. Since (n' v_N) > (v_n_i), there exists j > N such that v_j |_M m. Therefore
	
    Nv_N > n'v_N =  v_n_i + m  > m≥v_j≥v_N+1,

	which is a contradiction. Hence M is not weakly finitary.









 §.§ Strongly Primary Monoids


We conclude this paper with a few words about strongly primary monoids in 𝒞. Recall that a monoid is strongly primary if it is simultaneously finitary and primary. Numerical monoids and v-Noetherian primary monoids are strongly primary. On the other hand, the multiplicative monoid of a 1-dimensional local Mori domain is strongly primary. Finally, the class of strongly primary monoids also contains the class of finitely primary monoids <cit.>. Strongly primary monoids have been investigated in <cit.>.

Let M be a monoid. For x ∈ M^∙ the smallest n ∈ satisfying that nM^∙⊆ x + M is denoted by ℳ(x). When such n does not exist, we set ℳ(x) = ∞. If M is strongly primary, then ℳ(x) < ∞ for all x ∈ M^∙ <cit.>. In addition, set

    ℳ(M) := sup{ℳ(a) | a ∈𝒜(M) }⊆∪{∞}.


 
	Consider the monoid
	
    M := {(0,0)}∪{(x,y) ∈^2 | x,y > 0}.

	It is clear that
	
    A := {(a,b) ∈ M | a=1   or  b=1 }⊆𝒜(M).

	On the other hand, if (x,y) ∈ M^∙∖ A, then x,y ≥ 2 and, therefore,
	
    (x,y) = (1,1) + (x-1,y-1) ∈ M^∙ + M^∙.

	Hence 𝒜(M) = A. In addition, the fact that (1,1) |_M (x,y) for all (x,y) ∈ M^∙∖ A implies that ℳ((1,1)) = 2. The inclusion 2M^∙⊆ (1,1) + M implies that M is a finitary monoid. On the other hand, _^2(M)^∙ is the open first quadrant, which implies via Proposition <ref> that M is a primary monoid. As a result, M is strongly primary. Now fix n ∈_≥ 2. Note that if (n,1) |_M m(1,1) for some m ∈, then m ≥ n+1. Thus, ℳ((n,1)) ≥ n+1. On the other hand, if (x,y) ∈ (n+1) M^∙, then x ≥ n+1 and y ≥ 2, which implies that (x,y) - (n,1) ∈ M. As a result, ℳ((n,1)) = n+1 and, by a similar argument, ℳ((1,n)) = n+1. Hence ℳ((a,b)) = a+b for every (a,b) ∈ A and, in particular, ℳ(M) = ∞.


Unlike the computations shown in Example <ref>, an explicit computation of the set {ℳ(a) | a ∈ℳ} for a monoid M in 𝒞 can be hard to carry out. However, for most monoids M in 𝒞 one can argue that ℳ(M) = ∞ without performing such computations.


	Let M be a strongly primary monoid in 𝒞, and set V = ⊗_(M). The following statements are equivalent.
	
		
  (a) The inequality ℳ(M) < ∞ holds.
		
  (b) The equality _V(M) = 1 holds.
		
  (c) The monoid M is isomorphic to a numerical monoid.
	



	As a result of Proposition <ref> and Proposition <ref>, one can assume that M is a submonoid of (^d,+), where d is the rank of M. Then V = ^d.
	(b) ⇔ (c): It is clear.
	(a) ⇒ (b): To argue this suppose, by way of contradiction, that _V(M) ≠ 1. Since M is strongly primary M^∙ is not empty and, thus, _V(M) ≥ 2. As M is primary, _V(M)^∙ is an open set of V by Proposition <ref>. Therefore M cannot be finitely generated, which means that |𝒜(M)| = ∞. Since {a ∈𝒜(M) |a < n } is a finite set for every n ∈, there exists a sequence (a_n)_n ∈ of atoms of M satisfying that lim_n →∞a_n = ∞. Now fix x ∈ M^∙. Because ℳ(a_n)x = a_n + b for some b ∈ M,
	
    lim_n →∞ℳ(a_n) = lim_n →∞a_n + b/x≥1/xlim_n →∞a_n = ∞.

	Hence ℳ(M) = ∞, which is a contradiction.
	(b) ⇒ (a): To argue this, suppose that _V(M) = 1. In this case, M is isomorphic to a numerical monoid. Since numerical monoids are finitely generated, the inequality ℳ(M) < ∞ holds.










§ ACKNOWLEDGMENTS


While working on this paper, the author was supported by the NSF AGEP and the UC Year Dissertation Fellowship. The author would like to thank an anonymous referee for her/his useful comments.







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