TAME CONGRUENCES

Abstract

This chapter discusses a comprehensive theory of minimal congruences and of tame in finite algebras. Tame congruence theory reveals some unsuspected and important facets of finite algebras, and has a wide range of applications to all finite algebras and locally finite varieties. By A quotient in an algebra simply means a pair of congruences < α, β > and α < β. A quotient is called prime if α < β, that is, if α is covered by β. An algebra A is called < α,β >-minimal if A is an < α, β >-minimal set. Thus, a finite algebra A is minimal in our sense if every unary polynomial of A is a permutation or constant. A is ß-minimal if β is a nonzero congruence of A such that every unary polynomial is either a permutation or constant on each equivalence class of β.