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On Finite Presentability of Direct Products of Semigroups

On Finite Presentability of Direct Products of Semigroups A finite semigroup S is said to preserve finite generation (resp., presentability) in direct products, provided that, for every infinite semigroup T, the direct product S × T is finitely generated (resp., finitely presented) if and only if T is finitely generated (resp., finitely presented). The main result of this paper is a constructive necessary and sufficient condition for S to preserve both finite generation and presentability in direct products. The condition is that certain graphs, (s), one for each s ɛ S, are all connected. The main result is illustrated in three examples, one of which exhibits a 4-element semigroup that preserves finite generation but not finite presentability in direct products. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Colloquium Springer Journals

On Finite Presentability of Direct Products of Semigroups

Araújo, Isabel; Ruškuc, N.
Algebra Colloquium , Volume 7 (1) – Jan 1, 2000
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Publisher
Springer Journals
Copyright
Copyright © 2000 by Springer-Verlag Hong Kong
Subject
Mathematics; Algebra; Algebraic Geometry
ISSN
1005-3867
eISSN
0219-1733
DOI
10.1007/s10011-000-0083-4
Publisher site
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Abstract

A finite semigroup S is said to preserve finite generation (resp., presentability) in direct products, provided that, for every infinite semigroup T, the direct product S × T is finitely generated (resp., finitely presented) if and only if T is finitely generated (resp., finitely presented). The main result of this paper is a constructive necessary and sufficient condition for S to preserve both finite generation and presentability in direct products. The condition is that certain graphs, (s), one for each s ɛ S, are all connected. The main result is illustrated in three examples, one of which exhibits a 4-element semigroup that preserves finite generation but not finite presentability in direct products.

Journal

Algebra Colloquium – Springer Journals

Published: Jan 1, 2000

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