On the set of catenary degrees of finitely generated cancellative commutative monoids

Abstract

The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of n. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary degrees achieved by elements of S, focusing on the case where S in finitely generated (where C(S) is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of C(S) that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for C(S), and present some open questions for further study.

Factoring in the Chicken McNugget Monoid

“…Among these elements, 6,9,12,15,20,21,26,29,32,40,46, 49, and 52 each have a single factorization, and thus need not be considered. The factorizations of the remaining elements can each be found in Table 2 or 3.…”

Realisable Sets of Catenary Degrees of Numerical Monoids

“…In this section, we apply Theorem 3.3 to characterise the finite subsets of Z ≥0 which are realised as the set of catenary degrees of a numerical monoid, thus providing an alternative answer to [11,Problem 4.1] from that appearing in [5]. Proof.…”

“…For the converse direction, fix a finite set C satisfying conditions (i), (ii) and (iii). We will inductively build a monoid with set of catenary degrees C. If C = {0, c}, then C is the set of catenary degrees of c + 1, 2c + 1 by [11,Remark 4.2], and if C = {0, 2, c}, then C is the set of catenary degrees of 3, 3 + (c − 2), 3 + 2(c − 2) by [11,Theorem 4.3]. In all other cases, C = C ∩ [0, c) satisfies (i), (ii) and (iii) above, so we can inductively assume that C is the set of catenary degrees of some numerical monoid S = n 1 , .…”

“…In a similar vein, the catenary degree realisation problem [11,Problem 4.1] asks which finite sets can occur as the set C(M) of catenary degrees achieved by elements c 2017 Australian Mathematical Publishing Association Inc. 0004-9727/2017 $16.00 of some atomic monoid M. A recent paper by Fan and Geroldinger [5] proves that, with minimal assumptions, the product M = M 1 × M 2 of two monoids M 1 and M 2 has set of catenary degrees C(M) = C(M 1 ) ∪ C(M 2 ). While this does provide a complete solution to the catenary degree realisation problem, the brevity of the proof raises the question 'is such a result possible without appealing to Cartesian products?…”

Betti elements and catenary degree of telescopic numerical semigroup families

“…Let S be numerical semigroup and a ∈ S, x, y ∈ Z(a) and N ∈ N. In this case, the catenary degree of a, denoted by c(a), is the smallest of the existing N -chains. Furthermore, the set of catenary degrees of S is the set C(S) = {c(s) : s ∈ S}, and the catenary degree of S is the supremum of this set, namely c(S) = sup C(S) [1,3,7].…”