Park's conjecture

Abstract

A finite algebra of finite type (i.e. in a finite language) is finitely based iff the variety it generates can be axiomatized by finitely many equations. Park's conjecture states that if a finite algebra of finite type generates a variety in which all subdirectly irreducible members are finite and of bounded size, then the algebra is finitely based. In this thesis, I reproduce some of the finite basis results of this millennium, and give a taster of older ones. The main results fall into two categories: applications of Jonsson's theorem from 1979 (Baker's theorem in the congruence distributive setting, and its extension by Willard to congruence meet-semidistributive varieties), whilst other proofs are syntactical in nature (Lyndon's theorem on two element algebras, Je·zek's on poor signatures, Perkins's on commutative semigroups and the theorem on regularisation). The text is self-contained, assuming only basic knowledge of logic and universal algebra, and stating the results we build upon without proof.