Elastic Properties and Prime Elements

In a commutative, cancellative, atomic monoid M , the elasticity of a non-unit x is defined to be ρ(x) = L(x)/l(x), where L(x) is the supremum of the lengths of factorizations of x into irreducibles and l(x) is the corresponding infimum. The elasticity ρ(M) of M is given as the supremum of the elasticities of the nonzero non-units in the domain. We call ρ(M) accepted if there exists a non-unit x ∈ M with ρ(M) = ρ(x). In this paper, we show for a monoid M with accepted elasticity that {ρ(x) | x a non-unit of M} = Q ∩ [1, ρ(M)] if M has a prime element. We develop the ideas of taut and flexible elements to study the set {ρ(x) | x a non-unit of M} when M does not possess a prime element. Mathematics Subject Classification (2000). 20M14, 13F20, 13F15.