Three–element Nonfinitely Axiomatizable Matrices

In [2] W. Rautenberg proved that (the content of) any 2–element matrix is finitely axiomatizable and asked if the same is true for any finite matrix. This conjecture was disproved by P. Wojtylak, who in [5] (see also [4]) costructed a 5–element matrix with 2 designated elements, which is not finitely axiomatizable. Recently, W. Dziobiak presented in [1] a 4–element algebraic matrix (i.e. a matrix with one designated element) with the same property. His example and Rautenberg’s result mentioned above led Dziobiak to a natural question (also suggested by Wojtylak in [5]) whether nonfinitely axiomatizable matrices exist among the 3–element ones. In this note we give two such examples, both of which are algebraic. We also show that up to isomorphism these are the only nonfinitely axiomatizable 3– element algebraic matrices with one binary operation ◦ of which x◦(y◦z) is a tautology. This term is also a tautology of both Wojtylak’s and Dziobak’s matrices and plays an essential role in their proofs and also in the ours. We know nothing about the existence of nonfinitely axiomatizable matrices for which neither x◦(y◦z) nor (x◦y)◦z is a tautology. In our proof of nonfinite axiomatizability we are using the method of [4], [5] and [1] and we borrow some formulations from [1], [5] and [7]. Our language Λ is determined by an infinite set of variables