The set of minimal distances in Krull monoids

Abstract

Let H be a Krull monoid with finite class group G. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set L(a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ N such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ ∆ * (H), where ∆ * (H) denotes the set of minimal distances of H. We show that max ∆ * (H) ≤ max{exp(G) − 2, r(G) − 1} and that equality holds if every class of G contains a prime divisor, which holds true for holomorphy rings in global fields.

On congruence half-factorial Krull monoids with cyclic class group

“…If G is a finite cyclic group, by a result of Geroldinger and Hamidoune [20] (for earlier results see [12] and [15]), we have That is, also the second largest element is known; for results along these lines for more general groups we refer to [24,25,36]. Moreover, the structure of all sets corresponding to the maximal element |G| − 2 is known; this was proved in [35] for a larger class of groups, not only cyclic ones.…”

“…Since the introduction of these ideas, both these questions were investigated by various researchers (for recent contributions see, e.g., [4,12,20,24,25,34,44]). The investigations so far mainly focused on the first type of questions, due to the fact that a (partial) solution to it is a precondition for even beginning to consider the second one; see [24,35] for some initial results.…”

Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group

“…Moreover, directly from the above results, for exp(G) − 2 = r(G) the set ∆ * (G) must still be an interval, namely [1, exp(G)−2], yet for groups with r(G) < exp(G)− 2 the set ∆ * (G) could have gaps. Indeed, it frequently does have gaps, as the result below shows (it is a direct consequence of [41, Theorem 3.2] and [30]). Theorem 6.16.…”

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.…”