On the delta set and the Betti elements of a BF-monoid

We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of Δ(S) and the Betti elements of S. We prove how the minimum and maximum element of Δ(S) can be determined using the Betti elements of S. This leads to a determination of when Δ(S) is a singleton. We then apply these results to the particular case where S is a numerical monoid that requires three generators. Conclusions are drawn in the cases where S has a unique minimal presentation, or has multiplicity three.


Introduction
Recently, several papers describing the invariants of non-unique factorizations for finitely generated commutative cancellative monoids have appeared in the literature. Many of these papers have a special emphasis on numerical monoids. For example, [8] examined properties related to the elasticity of a numerical monoid. Moreover, the papers [2,5,[9][10][11] all consider properties related to the Delta set (or (S)) of a numerical monoid S. In particular, the results of [5] indicate that even if S is 3-generated, then the structure of (S) may be extremely complex (see for example, [5,Corollary 4.8 and Proposition 4.9]). In this note, we prove some general structure theorems for (S) when S is a reduced BF-monoid. Recall that S is a reduced BF-monoid if the only unit in S is its identity element, S is commutative and cancellative, and for every s ∈ S the set of lengths of its possible factorizations is bounded. We note that the reduced assumption is not necessary, since one can always remove the units of S and consider S red instead of S (see [14] for details). We then apply these results specifically to the case where S is a numerical monoid that requires three generators. This is reasonable, as several papers relating to algebraic properties of 3-generated numerical monoids have recently appeared in the literature (see [21,Chapter 9] and the references therein).
We present our results in three additional sections. In Sect. 2, we prove two structure theorems for the Delta set, and show connection between the elements of (S) and the Betti elements of S. We give explicit formulas for the minimum and maximum of (S). Thus we are able to determine when (S) is a singleton. Conclusions are drawn in Sect. 3 in the case where S has a unique minimal presentation. We then apply these results in Sect. 4 to the particular case where S is a 3-generated numerical monoid. We consider the pseudo-symmetric, symmetric, and multiplicity 3 cases. We believe our results are of interest, as they not only improve several results from [5], but they approach these problems in a much different manner using Betti elements. We open with some notation and definitions. Let N 0 represent the natural numbers including 0. Throughout our work, we assume that all monoids are commutative, cancellative, reduced, atomic, and the set of lengths of the factorizations of every element is bounded. For known results concerning non-unique factorizations in such monoids, the interested reader is directed to the monograph [14]. For such a monoid S, there exists a set of atoms, denoted A(S), such that every x ∈ S can be written in the form for some c 1 , . . . , c t ∈ N 0 , n 1 , . . . , n t ∈ A(S). We focus on the representations of elements of S in the form (*). Given x ∈ S with x = 0, set c a n a , c a ∈ N and n a ∈ A(S), for all a ⎫ ⎬ ⎭ which is known as the set of lengths of x in S. We are assuming that this set of lengths is bounded, and so L(x) is of the form {m 1 , m 2 , . . . , m k } for some positive integers m 1 < m 2 < · · · < m k . The set is known as the Delta set of x. We globalize the notion of the Delta set by setting By a fundamental result of Geroldinger [14,Proposition 1.4.4], if d = gcd (S) and | (S)| < ∞, then The study of Delta sets in the class of arithmetic congruence monoids can be found in [3]. A summary of known results involving properties of Delta sets can be found in [14,Section 6.7]. If S is an additive submonoid of N 0 , then S is called a numerical monoid. It follows from elementary number theory that S is finitely generated. If {n 1 , . . . , n e } is a set of generators for S, then this is commonly denoted by S = n 1 , . . . , n e . It also follows using basic techniques that the minimal generating set for S is unique. Unless otherwise noted, when dealing with a numerical monoid written as S = n 1 , . . . , n k , we assume that {n 1 , . . . , n k } is the minimal generating set for S. In a numerical monoid, 0 is the unique unit. A numerical monoid S = n 1 , . . . , n k is primitive if gcd{n 1 , . . . , n k } = 1. Clearly every numerical monoid is isomorphic to a primitive numerical monoid, hence we narrow our study to the primitive case (the interested reader is referred to [21] for more details on numerical monoids).
The following previously known results when S is a numerical monoid were the starting point of our work. Moreover, if S = n 1 , n 2 , n 3 is primitive and minimally generated, then [11,Theorem 3.1] provides a method for computing max (S) in terms of certain relations between the generators n 1 , n 2 and n 3 .

Betti elements and Delta sets
Let S be a BF-monoid, and let Z(S) = F(A(S)) the free monoid on the atoms of S. The unique monoid map π S : Z(S) → S that maps every a ∈ A(S) to a ∈ S is sometimes known as the factorization homomorphism associated to S. For every s ∈ S, the set Z(s) = π −1 S (s) is the set of factorizations of s. Let ∼ S be the kernel congruence of π S , i.e., x ∼ S y if π S (x) = π S (y), or equivalently, x and y are factorizations of the same element in S (∼ S is actually a congruence). It follows easily that S is isomorphic to the monoid Z(S)/ ∼ S .
For u = a∈A(S) u a a and v = a∈A(S) v a a ∈ Z(S), set gcd{u, v} = a∈A(S) min{u a , v a }a (this plays the role of the greatest common divisor, but with additive notation).
Given s ∈ S, we define the following binary relation on Z(s). For x, y ∈ Z(s), xRy if there exists a chain An element s ∈ S is said to be a Betti element if Z(s) has more than one R−class (see [12]). Observe that there are finitely many Betti elements in S if S is finitely presented. The set of Betti elements of S will be denoted by Betti(S). In the finitely generated case, Betti elements are tightly related to the Betti numbers of the minimal free resolution of the semigroup ring K [S], with K a field (see for instance [6]), and the elements used to construct a minimal presentation for S (we will address later).
Given x = a∈A(S) x a a a factorization of n ∈ S, set |x| = a∈A(S) x a . Clearly, if q(Z(S)) is the group generated by Z(S), we can extend |·| : q(Z(S)) → Z, and it is a linear map (Z denotes the set of integers). With this notation L(n) = {|x| : x ∈ Z(n)}. Recall that we are assuming that this set has finitely many elements.
The following result states that one can go from one factorization to another of the same element, just using the factorizations of the Betti elements. The idea of the proof is inspired by [20, Proposition 9.3].
Lemma 2.1 Let s ∈ S and x, y ∈ Z(s). Then there exists z 1 , . . . , z t ∈ Z(s) such that

) and a i not in the same R−class as b i (and thus a i and b i are factorizations of a Betti element of S).
Proof If x and y are not in the same R−class, then we are done. So assume that x and y are R related, and let us proceed by induction on max L(s) (if max L(s) = 1, then s is an atom, and x = y). By definition there exists By putting all these sequences together, we obtain the z 1 , . . . , z t of the statement.

Remark 2.2
The above lemma, and thus the whole paper, can be stated in the more general setting of monoids with the ascending chain condition for principal ideals (see [14,Definition 1.1.3]). This condition is equivalent to saying that every descending divisor sequence is stationary. Observe that s i "divides" s(s − s i ∈ S). Hence, we can repeat the process for s i and because of the descending divisor sequence condition, this procedure must end after a finite number of steps. Notice that by [14, Proposition 1.1.4] these monoids are always atomic.

Proposition 2.3 Let S be a BF-monoid and let s ∈ S.
For every x, y ∈ Z(s), |x| − |y| is of the form for some integers λ 1 , . . . , λ t , and δ i = |a i | − |b i | with a i and b i factorizations of a Betti element in different R−classes. In particular, every element in (S) is of this form.
Proof As x ∼ S y, by Lemma 2.1, there exits a chain for some c i ∈ Z(S) and a i , b i non-R-related factorizations of a Betti element. Thus and the proof follows easily. Thus d divides d , and this concludes the proof.

Theorem 2.5 If S is a BF-monoid with
Proof The inequality max n∈Betti(S) (max (n)) ≤ max (S) is clear, so let us focus on the other direction.
Assume to the contrary that max (S) > max (n) for all Betti elements n of S. As above, take x, y to be factorizations of an element s ∈ S so that |y| − |x| = max (S), and consequently no other factorization z of s fulfills |x| < |z| < |y|. As x ∼ S y, let x 1 , . . . , . . , z u , of n such that a i = z 1 , . . . , z u = b i , and |z j+1 |−|z j | ≤ max (n), which we are assuming to be smaller than max (S). But then z j + c i ∼ S x ∼ S y for all j, and from the choice of x and y, there is no j such that |x| < |z j + c i | < |y|. Again, we can find j ∈ {1, . . . , u − 1} such that |z j + c i | ≤ |x| < |y| ≤ |z j+1 + c i |. This leads to a contradiction, since max (S) = |y| − |x| ≤ |z j+1 + c i | − |z j + c i | = |z j+1 − z j | ≤ max (n) < max (S).
With this Theorem we get an easy characterization of finitely generated monoids with a singleton set of distances.

Corollary 2.6 The set (S) = ∅ is a singleton if and only if n∈Betti(S) (n) is a singleton.
Observe that numerical monoids are never half factorial (the set of lengths of its elements are singletons), hence there is always a minimal relation (a, b) ∈ σ with |a| = |b|, and so (π S (a)) is not empty.

Finitely generated monoids having a unique minimal presentation
Let S be a monoid minimally generated by {n 1 , . . . , n e }. Given σ ⊆ Z(S) × Z(S), the congruence generated by σ is the least congruence containing σ . If ∼ S is the congruence generated by σ , then we say that σ is a presentation of S. Rédei's theorem (see [18]) precisely states that every finitely generated monoid is finitely presented. A presentation for S is a minimal presentation if none of its proper subsets generates ∼ S . In our setting, all minimal presentations have the same cardinality. Next we briefly describe a procedure for finding all minimal presentations for S as presented in [20,Chapter 9].
For every s ∈ S, define σ s in the following way.
• If Z(s) has one R−class, then set σ s = ∅. • Otherwise, let R 1 , . . . , R k be the different R−classes of Z(s). Choose any a i ∈ R i for all i ∈ {1, . . . , k} and set σ s to be any set of k−1 pairs of elements in V = {a 1 , . . . , a k } so that any two elements in V are connected by a sequence of pairs in σ s (or their symmetrics). For instance, we can choose σ s = {(a 1 , a 2 ), . . . , (a 1 , a k )}.
Then σ = s∈S σ s is a minimal presentation of S. Moreover, in this way one can construct all minimal presentations for S. We say that S has a unique minimal presentation if for any other minimal presentation σ , and any relator (a, b) ∈ σ , then either (a, b) ∈ σ or (b, a) ∈ σ (i.e., σ is unique up to rearrangement of the components of its relators). Monoids having a unique minimal presentation have drawn the attention of many researchers in the last decade (see for instance [12]). {(a 1 , b 1 ), . . . , (a t , b t  We now recall the definition of another invariant that is tightly related to max (S). If s ∈ S, z, z ∈ Z(s), and N is a non-negative integer, then an N -chain of factorizations from z to z is a sequence z 0 , . . . , z k ∈ Z(s) such that z 0 = z, z k = z and max{|z i − d i |, |z i+1 − d i |) ≤ N for all i, and d i = gcd(z i , z i+1 ). The catenary degree of S, c(S), is the minimal N ∈ N 0 ∪{∞} such that for any s ∈ S and any two factorizations z, z ∈ Z(s), there is an N -chain from z to z . It is well known ([14, Proposition 1.6.3]) that

Corollary 3.1 Assume that S is uniquely presented with (S) = ∅, and that its unique minimal presentation is
Observe also that if S is a uniquely presented finitely generated monoid, then as a consequence of [7, Theorem 3.1] c(S) = max{max{|a i |, |b i |} : i ∈ {1, . . . , t}}.

Embedding dimension three numerical monoids
Let us recall some basic facts concerning numerical monoids in general. Define The cardinality of this set is the type of S. Observe that the largest integer not belonging to S, its Frobenius number denoted by F(S), is always in T(S) (recall that we are assuming that S is primitive, and thus N \ S is always finite). Numerical monoids of type one are symmetric (or equivalently, the cardinality of N \ S equals F(S)+1 2 ; see [21] for more details), and numerical monoids with T(S) = {F(S), F(S)/2} are called pseudosymmetric. For an element n ∈ S, define its Apéry set as Ap(S, n) = {s ∈ S | s − n ∈ S}. This set contains exactly n elements, each being the minimum in its congruence class modulo n.
We now restrict to embedding dimension three numerical monoids. In this setting, minimal presentations either have cardinality two or three, and the shape of these presentations are well known (see for instance [21,Chapter 9]). Numerical monoids having just a couple of relations are complete intersections (both free and symmetric). We show that the sets of distances can be easily described in this setting.
With this we obtain the following.
A minimal presentation for S (uniqueness is only granted when 0 < b < m 2 and 0 < c < m 1 , see [12,Theorem 17]) is