Externalization of lattices

• Autorzy: Chajda I. , Wismath S.
• Języki publikacji: EN

Abstrakty

EN

Let r be a type of algebras. An identity s = t of type r is said to be externally compatible, or simply external, if the terms s and t are either the same variable or both start with the same operation symbol fj of the type. A variety is called external if all of its identities are external. For any variety V , there is a least external variety E(V ) containing V , the variety determined by the set of all external identities of V . External identities and varieties have been studied by [4], [5] and [2], and a general characterization of the algebras in E(V ) has been given in [3]. In this paper we study the algebras of the variety E(V ) where V is the type (2, 2) variety L of lattices. Algebras in L may also be described as ordered sets, and we give an ordered set description of the algebras in E(L). We show that on any algebra in E(L) there is a natural quasiorder having an additional property called externality, and that any set with such a quasiorder can be given the structure of an algebra in E(L). We also characterize algebras in E(L) by an inflation construction.

Słowa kluczowe

EN

externally compatible identity; lattice; externalization of lattices

PL

kraty; eksternalizacja krat

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2006
• Tom: Vol. 39, nr 4
• Strony: 731--736
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Chajda I.

autor

Wismath S.

• Department Algebra and Geometry Palacky University Tomkova 40, 779 00 Olomouc, Czech Republic,

Bibliografia

• [1] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327–335.
• [2] W. Chromik, Externally compatible identities of algebras, Demonstratio Math. 23 (1990), no. 2, 345–355.
• [3] I. Chajda, K. Denecke and S. L. Wismath, A characterization of P-compatible varieties, to appear in Algebra Colloquium.
• [4] J. Płonka, On varieties of algebras defined by identities of special forms, Houston Mathem. J. 14 (1988), 253–263.
• [5] J. Płonka, P-compatible identities and their applications to classical algebras, Math. Slovaca 40 (1990), no. 1, 21–30.