FULL ELASTICITY IN ATOMIC MONOIDS AND INTEGRAL DOMAINS

Let M be a commutative cancellative atomic monoid and M* its set of nonunits. Let ρ(x) denote the elasticity of factorization of x ϵ M*, R(M) = {ρ(x) | x ϵ M*} the set of elasticities of elements of M, and ρ(M) = sup R(M) the elasticity of M. We say M is fully elastic if R(M) = Q ⋂ [1, ρ(M)]. We cal...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2006-01, Vol.36 (5), p.1437-1455
Main Authors: CHAPMAN, SCOTT T., HOLDEN, MATTHEW T., MOORE, TERRI A.
Format: Article
Language:English
Subjects:
11Y05
13F05
20M14
20M25
Algebra
Factorials
Factorization
Integers
Mathematical integrals
Mathematical rings
Mathematical sets
Mathematical theorems
Mathematics
Monoids
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Summary:Let M be a commutative cancellative atomic monoid and M* its set of nonunits. Let ρ(x) denote the elasticity of factorization of x ϵ M*, R(M) = {ρ(x) | x ϵ M*} the set of elasticities of elements of M, and ρ(M) = sup R(M) the elasticity of M. We say M is fully elastic if R(M) = Q ⋂ [1, ρ(M)]. We call an atomic integral domain D fully elastic if its multiplicative monoid, denoted D*, is fully elastic. We examine the full elasticity property in the context of Krull monoids with finite divisor class groups, numerical monoids and certain integral domains. For every real number α ≥ 1, we construct a fully elastic Dedekind domain D with ρ(D) = α. In particular, while we show that noncyclic numerical monoids are never fully elastic, we do verify that several large classes of Krull monoids, and hence certain Krull domains, are fully elastic.
ISSN:0035-7596
1945-3795
DOI:10.1216/rmjm/1181069375