Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-04T20:29:34.030Z Has data issue: false hasContentIssue false

On equational theories of semilattices with operators

Published online by Cambridge University Press:  17 April 2009

J. Ježek
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Sokolovská 83, 180 00 Prague 8, Czechoslovakia
P. PudláK
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Sokolovská 83, 180 00 Prague 8, Czechoslovakia
J. Tůma
Affiliation:
Mathematical Institute, Žitná, 25 115 67 Prague 1, Czechoslovakia
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.

In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Freese, R. and Nation, J.B., ‘Congruence lattices of semilattices’, Pacific J. Math. 49 (1973), 5158.CrossRefGoogle Scholar
[2]Lampe, W., ‘A property of the lattice of equational theories’, Algebra Universalis 23 (1986), 6169.CrossRefGoogle Scholar
[3]Lampe, W., ‘Further properties of lattices of equational theories’, (preprint).Google Scholar
[4]Papert, D., ‘Congruence relations in semi-lattices’, J. London Math. Soc. 39 (1964), 723729.CrossRefGoogle Scholar
[5]Pigozzi, D., ‘The representation of certain abstract lattices as lattices of subvarieties’, (preprint).Google Scholar
[6]Zhitomirskii^, G.I., O reshetke vsekh kongruèntsii^ polureshetki (On the lattice of all congruences of a semilattice): Uporyadochennye mnozhestva i reshetki No.l, (Ordered sets and lattices), pp. 1121 (Saratovskii^ gos. un-t, 1971).Google Scholar