Accepted Elasticity in Local Arithmetic Congruence Monoids

Abstract

For certain ${a,b \in \mathbb{N}}$, an Arithmetic Congruence Monoid M(a, b) is a multiplicatively closed subset of ${\mathbb{N}}$ given by ${\{x\in\mathbb{N}:x \equiv a \pmod{b}\} \cup\{1\}}$. An irreducible in this monoid is any element that cannot be factored into two elements, each greater than 1. Each monoid element (apart from 1) may be factored into irreducibles in at least one way. The elasticity of a monoid element (apart from 1) is the longest length of a factorization into irreducibles, divided by the shortest length of a factorization into irreducibles. The elasticity of the monoid is the supremum of the elasticities of the monoid elements. A monoid has accepted elasticity if there is some monoid element that has the same elasticity as the monoid. An Arithmetic Congruence Monoid is local if gcd(a, b) is a prime power (apart from 1). It has already been determined whether Arithmetic Congruence Monoids have accepted elasticity in the non-local case; we make make significant progress in the local case, i.e. for many values of a, b.