The Equational Theory of Paramedial Cancellation Groupoids

By a paramedial groupoid we mean a groupoid satisfying the equation xy · zu = uy · zx. As it is easy to see, the equational theory of paramedial groupoids, as well as the equational theory based on any balanced equation, is decidable. In this paper we are going to investigate the equational theory of paramedial cancellation groupoids; by this we mean the set of all equations satisfied by paramedial cancellation groupoids. (By a cancellation groupoid we mean a groupoid satisfying both xz = yz → x = y and zx = zy → x = y.) Clearly, the equational theory of paramedial cancellation groupoids is just the least cancellative equational theory containing the paramedial law. We will show that this equational theory is also decidable (Theorem 4.1), that it is a proper extension of the equational theory of paramedial groupoids (Theorem 4.3), and that whenever two terms are unrelated with respect to this equational theory, then their squares are also unrelated (Theorem 4.7). The results can be compared with those of [2] and [3] for medial groupoids.