On delta sets and their realizable subsets in Krull monoids with cyclic class groups

Abstract

Abstract. Let M be a commutative cancellative monoid. The set ∆(M ), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M , is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that ∆(M ) ⊆ {1, . . . , n − 2}. Moreover, equality holds for this containment when each class contains a prime divisor from M . In this note, we consider the question of determining which subsets of {1, . . . , n−2} occur as the delta set of an individual element from M . We first prove for x ∈ M that if n − 2 ∈ ∆(x), then ∆(x) = {n − 2} (i.e., not all subsets of {1, . . . , n − 2} can be realized as delta sets of individual elements). We close by proving an Archimedean-type property for delta sets from Krull monoids with finite cyclic class group: for every natural number m, there exist a Krull monoid M with finite cyclic class group such that M has an element x with |∆(x)| ≥ m.

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.…”

The Catenary Degree of Krull Monoids Ii

“…The present paper will focus on the three closely related invariants, namely the set of distances, the (H) The study of these arithmetical invariants (in settings ranging from numerical monoids to Mori rings with zero-divisors) has attracted a lot of attention in the recent literature (for a sample see [10,9,24,14,25,7,11,8]). Our main focus here will be on Krull monoids with finite class group G such that each class contains a prime divisor.…”

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

“…Recently, Chapman, Gotti, and Pelayo [6] obtained the following result on this type of problem. We recall that n − 2 is the maximum of the set of distances for Krull monoid with cyclic class group n, assuming that each class contains a prime divisor.…”