Amalgamation in small varieties of lattices
The amalgamation class Amal(\({\mathcal V})\) of the variety \({\mathcal V}\) was introduced by H. Lakser and the reviewer as the class of all algebras in \({\mathcal V}\) over which one can amalgamate in \({\mathcal V}\); thus, Amal(\({\mathcal V})\) measures to what extent the Amalgamation Property holds (fails) in \({\mathcal V}.\) The author's abstract: ``A property used by G. Grätzer and H. Lakser to describe the amalgamation class of a finitely generated variety of pseudo-complemented distributive lattices, and a property used by C. Bergman in his investigations of the amalgamation classes of varieties that are finitely generated, congruence distributive, and semi-simple, are here applied to arbitrary finitely generated, congruence distributive varieties. The Grätzer-Lakser property is shown to characterize the amalgamation class of a finitely generated class \({\mathcal V}\) of lattices. For finite lattices, this gives an effective test for membership in the amalgamation class of \({\mathcal V}\). For \({\mathcal V}={\mathcal N}\), the variety generated by the pentagon, an even simpler test is found.'' The last sentence of the author's abstract refers to the following elegant result: A finite lattice \(A\in {\mathcal N}\) belongs to Amal(\({\mathcal N})\) iff A is a subdirect power of N and A does not have the three element chain as a homomorphic image.
finitely generated varieties of lattices
scientific article; zbMATH DE number 4212117