Abstract

The question whether the consequence operation of a nite matrix is always nitely based was proposed by S. L. Bloom [1] and also by R. Wojcicki [3]. The negative answer { a six-element matrix { was given in [4] and next a ve-element matrix was found by A. Urquhart [2]. In this paper we will show that no nite basis exists for the consequence operation CM of the matrix M = hhf0; 1; 2g; i; f2gi where is a binary operation such that 0 1 = 0 and a b = 2 otherwise. Our proof is a slight modication of that given by A. Urquhart in [2] and most of the notions and techniques used here are borrowed from [2]. Our result, however, is the best possible because every consequence operation of a two element matrix is nitely based