Elasticity in certain block monoids via the Euclidean table |
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Authors: | SooAh Chang Scott T Chapman William W Smith |
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Institution: | (1) Department of Mathematics, The University of North Carolina at Chapel Hill, Phillips Hall, Chapel Hill, NC 27599-3250, USA;(2) Department of Mathematics, Trinity University, One Trinity Place, San Antonio, TX 78212-7200, USA |
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Abstract: | This paper continues the study begun in GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤ
n
, S) where S =
(hereafter denoted ℬ
a
(n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬ
a
(n). In Section 3 we use the Euclidean table to compute the elasticity of ℬ
a
(n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬ
a
(n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬ
a
(p).
Part of this work was completed while the second author was on an Academic Leave granted by the Trinity University Faculty
Development Committee. |
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Keywords: | block monoid elasticity of factorization non-unique factorization minimal zero-sequence |
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