Elastic Properties and Prime Elements

Autor: K. Grace Kennedy, Paul Baginski, Christopher Crutchfield, Scott T. Chapman, Matthew Wright
Rok vydání: 2006
Předmět:
Zdroj: Results in Mathematics. 49:187-200
ISSN: 1420-9012
1422-6383
DOI: 10.1007/s00025-006-0219-z
Popis: 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 $$ \{\rho (x)\,|\,x\,{\text{a non - unit of }}M\} = {\user2{\mathbb{Q}}} \cap [1,\rho (M)]{\user2{}} $$ 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.
Databáze: OpenAIRE