On the Delta set and catenary degree of Krull monoids with infinite cyclic divisor class group

Abstract

a b s t r a c t Let M be a Krull monoid with divisor class group Z, and let S ⊆ Z denote the set of divisor classes of M which contain prime divisors. We find conditions on S equivalent to the finiteness of both ∆(M), the Delta set of M, and c(M), the catenary degree of M. In the finite case, we obtain explicit upper bounds on max ∆(M) and c(M). Our methods generalize and complement a previous result concerning the elasticity of M.

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

“…If there are classes without prime divisors, then the knowledge on the arithmetic is still very limited. Chapman et al studied the arithmetic of a Krull monoid with infinite cyclic class group (see [6,7]; for more on the arithmetic in the case of infinite class groups see [28,19,21]). In Theorem 4.2 of the present paper, we prove that, for a large class of Krull monoids (including all Krull monoids with torsion class group and all Krull domains) the monoid is tame if and only if the associated Davenport constant is finite.…”

The set of distances in seminormal weakly Krull monoids

“…. + L(a) is contained in L(a n ) whence |L(a n )| > n for every n ∈ N. The set of distances ∆(H) (also called the delta set of H) is the union of all sets ∆(L(a)) over all non-units a ∈ H. The set of distances (together with associated invariants, such as the catenary degree) has found wide interest in the literature in settings ranging from numerical monoids to Mori domains (for a sample out of many see [11,9,4,15,16,10,8,12,21,30]). In the present paper we focus on seminormal weakly Krull monoids and show -under mild natural assumptions -that their sets of distances are intervals.…”

A note on minimal zero-sum sequences over Z

“…The study of zero-sum sequences in B(G), when G a finite cyclic group, is a very active area of research (e.g., see [2,5,6,9,18,19,22]) with applications to Factorization Theory (e.g., see [3,10,11,12]). Similar, but less extensive, investigations have been carried out when G is an infinite cyclic group (e.g., see [4,7,13,14]).…”

“…If S is as in (2), then |S + | ≤ S − ∞ = b m and |S − | ≤ S + ∞ = a n . This was reformulated and reproved in the language of sequences by Baginski et al [4]. Perhaps due to inconsistent notation across various areas, Theorem 1.1 has been independently rediscovered by Diaconis et al [8], and Sahs et al [21].…”