A1 Refereed original research article in a scientific journal
Compactness of systems of equations in semigroups
Authors: Harju T, Karhumaki J, Plandowski W
Publication year: 1995
Journal:: Lecture Notes in Computer Science
Journal name in source: AUTOMATA, LANGUAGES AND PROGRAMMING
Journal acronym: LECT NOTES COMPUT SC
Volume: 944
First page : 444
Last page: 454
Number of pages: 11
ISBN: 3-540-60084-1
ISSN: 0302-9743
Abstract
We consider systems u(i) = v(i) (i epsilon I) of equations in semigroups over finite sets of variables. A semigroup S is said to satisfy the compactness property (or CP, for short), if each system of equations has an equivalent finite subsystem. It is shown that all monoids in a variety V satisfy CP, if and only if the finitely generated monoids in V satisfy the maximal condition on congruences. We also show that if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. Finally, we give three simple examples (the bicyclic monoid, the free monogenic inverse semigroup and the Baumslag-Solitar group) which do not satisfy CP, and show that the above necessary conditions are not sufficient.
We consider systems u(i) = v(i) (i epsilon I) of equations in semigroups over finite sets of variables. A semigroup S is said to satisfy the compactness property (or CP, for short), if each system of equations has an equivalent finite subsystem. It is shown that all monoids in a variety V satisfy CP, if and only if the finitely generated monoids in V satisfy the maximal condition on congruences. We also show that if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. Finally, we give three simple examples (the bicyclic monoid, the free monogenic inverse semigroup and the Baumslag-Solitar group) which do not satisfy CP, and show that the above necessary conditions are not sufficient.
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