Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T06:07:28.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Kleene algebras are almost universal

Published online by Cambridge University Press:  17 April 2009

M. E. Adams  and
H. A. Priestley
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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 3 , December 1986 , pp. 343 - 373
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Adams, M. E., Koubek, V., and Sichler, J., “Homomorphisms and endomorphisms in varieties of pseudocomplemented distributive lattices (with applications to Heyting algebras)”, Trans. Amer. Math. Soc. 285 (1984), 57–79.CrossRefGoogle Scholar
[2]Adams, M. E., Koubek, V., and Sichler, J., “Homomorphisms and endomorphisms of distributive lattices”, Houston J. Math. 11 (1985), 129–145.Google Scholar
[3]Adams, M. E. and Priestley, H. A., “De Morgan algebras are universal”, (to appear).Google Scholar
[4]Balbes, R. and Dwinger, Ph., Distributive Lattices, (university of Missouri Press, Columbia, Missouri, 1974).Google Scholar
[5]Beazer, R., “On some small varieties of distributive Ockham algebras”, Glasgow Math. J. 25 (1984) 175–181.CrossRefGoogle Scholar
[6]Berman, J., “Distributive lattices with and additional unary operation” Aequationes Math. 16 (1977), 165–171.CrossRefGoogle Scholar
[7]Blyth, T. S. and Varlet, J. C., “On a common abstraction of de Morgan algebras and Stone algebras”, Proc. Roy. Soc. Edinburgh Sect.A 94 (1983), 301–308.CrossRefGoogle Scholar
[8]Blyth, T. S. and Varlet, J. C., “Subvarieties of the class of MS-algebras”, Proc. Roy. Soc. Edinburgh Sect.A 95 (1983), 157–169.CrossRefGoogle Scholar
[9]Clark, D. M. and Krauss, P. H., “On topological quasivarieties”, Acta Sci. Math. 47 (1984), 3–39.Google Scholar
[10]Cornish, W. H. and Fowler, P. R., “Coproducts of de Morgan algebras”, Bull. Austral. Math. Soc. 16 (1977), 1–13.CrossRefGoogle Scholar
[11]Cornish, W. H. and Fowler, P. R., “Coproducts of Kleene algebras”, J. Austral. Math. Soc. Ser. A 27 (1979), 209–220.CrossRefGoogle Scholar
[12]Davey, B. A. and Duffus, D., “Exponentiation and duality”, in Ordered Sets (ed. Rival, I.), NATO Advanced Study Institutes Series, D. Reidel, Dordrecht, 1982, pp. 43–96.CrossRefGoogle Scholar
[13]Davey, B. A. and Priestley, H. A., “Generalised piggyback dualities and applications to Ockham algebras”, Houston J. Math., (to appear).Google Scholar
[14]Davey, B. A. and Werner, H., “Dualities and equivalences for varieties of algebras”, in Contributions to Lattice Theory (Szeged 1980), Colloq. Math. Soc. János Bolyai 33, North-Holland, Amsterdam-New York, 1983, pp. 101–275.Google Scholar
[15]Fowler, P. R., De Morgan Algebras, (ph.D. Thesis, Flinders University, Australia, 1980.)Google Scholar
[16]Goldberg, M. S., Distributive p-algebras and Ockham Algebras: a Topological Approach, (ph.D. Thesis, La Trobe University, Australia, 1979).Google Scholar
[17]Goldberg, M. S., “Distributive Ockham algebras: free algebras and injectivity”, Bull. Austral. Math. Soc. 24 (1981), 161–203.CrossRefGoogle Scholar
[18]Grȁtzer, G., Lattice Theory: First Concepts and Distributive Lattices (Freeman, San Francisco, California, 1971).Google Scholar
[19]Hedrlín, Z. and Pultr, A., “Symmetric relations (undirected graphs) with given semigroup”, Monatsh. Math. 68 (1964), 421–425.CrossRefGoogle Scholar
[20]Hedrlín, Z. and Pultr, A., “On full embeddings of categories of algebras”, Illinois J. Math. 10 (1966), 392–406.CrossRefGoogle Scholar
[21]Kalman, J.A., “Lattices with involution”, Trans. Amer. Math. Soc. 87 (1958), 485–491.CrossRefGoogle Scholar
[22]Koubek, V., “Infinite image homomorphism of distributive bounded lattoces”, lectures in Universal Algebra (Szeged 1983), Colloq. Math. Soc. Janos Bolyai 43 North Holland, Amsterdam – New York, (1985) 241–281.Google Scholar
[23]Koubek, V. and Sichler, J., “Universal varieties of distributive double p-algebras”, Glasgow Math. J. 26 (1985), 121–131.CrossRefGoogle Scholar
[24]Magill, K. D., “The semigroup of endomorphisms of a Boolean ring”, Semigroup Forum 4 (1972), 411–416.Google Scholar
[25]Maxson, C. J., “On semigroups of Boolean ring endomorphisms”, Semigroup Forum 4 (1972), 78–82).CrossRefGoogle Scholar
[26]Priestley, H. A., “Representation of distributive lattices by means of ordered Stone spaces”, Bull. London Math. Soc. 2 (1970), 186–190.CrossRefGoogle Scholar
[27]Priestley, H. A., “Ordered topological spaces and the representation of distributive lattices”, Proc. London Math. Soc. (3) 24 (1972), 507–530.CrossRefGoogle Scholar
[28]Priestley, H. A., “Ordered sets and duality for distributive lattices”, Ann. Discrete Math. 23 (1984), 39–60.Google Scholar
[29]Pultr, A. and Trnková, V.Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, (North-Holland, Amsterdam, 1980.)Google Scholar
[30]Schein, B. M., “Ordered sets, semilattices, distributive lattices Boolean algebras with homomorphic endomorphism semigroups”, Fund. Math. 68 (1970), 31–50.CrossRefGoogle Scholar
[31]Urquhart, A., “Distributive lattices with a dual homomorphic operation”, Studia Logica 38 (1979), 201–209.CrossRefGoogle Scholar