On the Delta set of a singular arithmetical congruence monoid

A theorem on accepted elasticity in certain local arithmetical congruence monoids

“…Moreover, each B-type atom must have at least one element in [1] or [3] (as p k is in [3], any number with p-adic value k with no prime factor other than p in [1] or [3] must be in either [1] or [3] itself and therefore not in M(p k a, p k b)). However, each A-type atom can have no more than three prime factors in [3] (as the product of three factors in [3] is in [1]); therefore, each A-type atom has at most two prime factors in [3]. Now suppose W has the same elasticity as the monoid.…”

“…Each prime factor other than p which divides an A-type atom must be in [3], [2], or [0] (this holds as every prime in [1] has value 3). Moreover, each B-type atom must have at least one element in [1] or [3] (as p k is in [3], any number with p-adic value k with no prime factor other than p in [1] or [3] must be in either [1] or [3] itself and therefore not in M(p k a, p k b)).…”

“…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 [3,5] and [4]) and are further documented in the recent monograph [7]. ACMs belong to a larger category of monoids known as congruence monoids (see [6,8] and [9]).…”

On the Elasticity of Generalized Arithmetical Congruence Monoids

“…When gcd(a, b) = p k for p a prime integer, then M (a, b) is called local. Some results concerning the arithmetic of M (a, b) in the local case can be found in [2][3][4].…”

Bifurcus semigroups and rings

“…For a background to factorization theory, please see [5]. For additional undefined terms, please see [2,3].…”