A Theorem on Accepted Elasticity in Local Arithmetical Congruence Monoids

We give necessary and sufficient conditions for a certain class of local arithmetical congruence monoids (or ACMs) to have accepted elasticity. Let a and b be positive integers with a ≤ b and a ≡ a (mod b). Under these conditions, the arithmetic sequence a, a+ b, a+ 2b, a+ 3b, . . . is closed under multiplication and is a multiplicative semigroup. Setting M(a, b) = {a+ kb | k ∈ N0} ∪ {1} produces a monoid known as an arithmetical congruence monoid (or an ACM). ACMs belong to a larger category of monoids known as congruence monoids (see [6], [8] and [9]). The properties relating to non-unique factorizations of elements in ACMs into products of irreducible elements have been the subject of several recent papers (see [3], [4] and [5]) and are further documented in the recent monograph [7]. In this paper, we will explore when a certain class of ACMs has accepted elasticity. Before stating our main result, we will need a series of definitions and a brief review of the main results of the papers mentioned above. Suppose M(a, b) is an ACM. Set d = gcd (a, b) and m = b/d. M(a, b) is called regular if d = 1 and singular if d > 1. By [5, Proposition 2.2], regular ACMs are Krull monoids (see [7, Section 2.5]) whose arithmetic is well documented. Hence, in this paper, we focus on the singular case. In the singular case, it follows from [3, Theorem 2.1] that gcd(a,m) = gcd(b,m) = gcd(d,m) = 1. If a singular ACM M(a, b) has d = p for some prime integer p, then M(a, b) is called local (this terminology is used since any singular ACM can be decomposed as an intersection of local ACMs [3, Section 4]). By the proof of [5, Theorem 2.4], if M(a, b) is a local ACM with d = p, then there exists a minimal positive integer n such that p ∈M(a, b). 1991 Mathematics Subject Classification. 20M14, 20D60, 13F05.