Laws in Finite Algebras

Abstract

Publisher Summary This chapter discusses certain classes of finite algebras that have finite bases for their laws. It also provides examples of finite algebras that do not have finite bases for their laws. It is shown that if A is a finite algebra then the variety generated by A , Var A , is a locally finite variety. Any locally finite variety is generated by its critical algebras and so the critical algebras in Var A play an important role. A locally finite variety V is Cross if (a) V has a finite basis for its laws, and (b) V has only finitely many critical algebras. Finite groups, rings, Lie rings, and finite algebras in varieties all of whose algebras have distributive congruence lattices, all generate Cross varieties. The basic method of proving that a finite algebra A generates a Cross variety is to find a finite set of first order sentences with the following properties: (1) every algebra in Var A satisfies finite set,(2) every finitely generated algebra satisfying finite set is finite, and (3) there are only finitely many critical algebras satisfying finite set.