Finite basis problems and results for quasivarieties

Let K be a finite collection of finite algebras of finite signature such that SP( K ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection K 1 of finite algebras of the same signature, K1⊇K , such that SP( K 1) is finitely axiomatizable.We show also that if HS(K)⊆SP(K) , then SP( K 1) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates