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Polynomial density of commutative semigroups

Published online by Cambridge University Press:  17 April 2009

Andrzej Kisielewicz  and
Norbert Newrly
Andrzej Kisielewicz
Affiliation:
Institute of Mathematics Technical University of Wroclaw, Wybrzeże Wyspiańskiego 27 50-370 Wroclaw, Polandkisiel@plwrtull.bitnet
Norbert Newrly
Affiliation:
Technische Hochschule Darmstadt Fachbereich 4 Schlossgartenstr. 7 6100 Darmstadt, Germanyxmatag01@ddathd21.bitnet
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An algebra is said to be polynomially n−dense if all equational theories extending the equational theory of the algebra with constants have a relative base consisting of equations in no more than n variables. In this paper, we investigate polynomial density of commutative semigroups. In particular, we prove that, for n > 1, a commutative semigroup is (n − 1)-dense if and only if its subsemigroup consisting of all n−factor-products is either a monoid or a union of groups of a bounded order. Moreover, a commutative semigroup is 0-dense if and only if it is a bounded semilattice. For semilattices, we give a full description of the corresponding lattices of equational theories.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 1 , August 1993 , pp. 151 - 162
Copyright
Copyright © Australian Mathematical Society 1993

References

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