Externalization of lattices
Wybrane pełne teksty z tego czasopisma
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 ,  and , and a general characterization of the algebras in E(V ) has been given in . 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.
Bibliogr. 5 poz.
-  I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327–335.
-  W. Chromik, Externally compatible identities of algebras, Demonstratio Math. 23 (1990), no. 2, 345–355.
-  I. Chajda, K. Denecke and S. L. Wismath, A characterization of P-compatible varieties, to appear in Algebra Colloquium.
-  J. Płonka, On varieties of algebras defined by identities of special forms, Houston Mathem. J. 14 (1988), 253–263.
-  J. Płonka, P-compatible identities and their applications to classical algebras, Math. Slovaca 40 (1990), no. 1, 21–30.