We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.

We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.

There are exactly two nonfinitely axiomatizable algebraic matrices with one binary connective o such thatx(yz) is a tautology of . This answers a question asked by W. Rautenberg in [2], P. Wojtylak in [8] and W. Dziobiak in [1]. Since every 2-element matrix can be finitely axiomatized ([3]), the matrices presented here are of the smallest possible size and in some sense are the simplest possible.

Motivated by a question of W. Rautenberg, we prove that any matrix that is term-equivalent to the well-known nonfinitely based matrix of A. Wroski is itself also nonfinitely based.

It was shown in [5] that all two-element matrices are finitely based independently of their classification by term equivalence. In particular, each 2-valued matrix is finitely axiomatizable. We show below that for certain two not finitely axiomatizable 3-valued matrices this property is also preserved under term equivalence. The general problem, whether finite axiomatizability of a finite matrix is preserved under term-equivalence, is still open, as well as the related problem as to whether the consequence operation of a finite matrix is (...) finitely based. (shrink)