Elasticity in certain block monoids via the Euclidean table

Abstract

Abstract This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤn, S) where S = $$\{ \bar 1,\bar a\} $$ (hereafter denoted ℬa(n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬa(n). In Section 3 we use the Euclidean table to compute the elasticity of ℬa(n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬa(n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬa(p).