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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: some proofs are applications of Jónsson'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, Ježek'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.
Keywords:finite basis, variety, finite residual bound
The text contains full proofs of the following theorems:- Birkhoff: variety which has a base in n variables
- Lyndon: two-element algebras are finitely based (only sketch of the proof)
- Shapiro: polynomial free spectrum
- Ježek: poor signatures
- regularisation of a finitely based variety
- Perkins: commutative semigroups
- Jónsson: very general finite basis theorem
- McKenzie: definable principal congruences
- definable disjointness of principal congruences
- Baker and Wang: definable principal subcongruences
- Baker: congruence-distributive algebras (two proofs)
- Willard: congruence meet semi-distributive algebras (two proofs)
Note: these are not full statements of the theorems, additional assumptions may be needed.