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Kleene algebras are almost universal

Published online by Cambridge University Press:  17 April 2009

M. E. Adams
Affiliation:
State University of New York, New Paltz, New York 12561, U.S.A.
H. A. Priestley
Affiliation:
Mathematical Institute, 24/29 St. Giles, Oxford OX1 3LB, England.
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Abstract

This paper studies endomorphism monoids of Kleene algebras. The main result is that these algebras form an almost universal variety k, from which it follows that for a given monoid M there is a proper class of non-isomorphic Kleene algebras with endomorphism monoid M+ (where M+ denotes the extension of M by a single element that is a right zero in M+). Kleene algebras form a subvariety of de Morgan algebras containing Boolean algebras. Previously it has been shown the latter are uniquely determined by their endomorphisms, while the former constitute a universal variety, containing, in particular, arbitrarily large finite rigid algebras. Non-trivial algebras in K always have non-trivial endomorphisms (so that universality of K is ruled out) and unlike the situation for de Morgan algebras the size of End(L) for a finite Kleene algebra L necessarily increases as |L| does. The paper concludes with results on endomorphism monoids of algebras in subvarieties of the variety of MS-algebras.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1986

References

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