A basis for the type 〈n〉 hyperidentities of the medial variety of semigroups

Abstract

The medical varietyMV of semigroups is the variety defined by the medial identityxyzw = xzyw. This variety is known to satisfy the medial hyperidentitiesF(G(x 11 ,⋯, x 1n ),⋯, G(x n1 ,⋯, x nn )) = G(F(x 11 ,⋯, x n1 ),⋯, F(x 1n ,⋯, x nn )), forn ≥ 1. Taylor has observed in [2] thatMV also satisfies some other hyperidentities, which are not consequences of the medial ones. In [4] the author introduced a countably infinite family of binary hyperidentities called transposition hyperidentities, which are natural generalizations of then = 2 medial hyperidentity. It was shown that this family is irredundant, and that no finite basis is possible for theMV hyperidentities with one binary operation symbol. In this paper, we generalize the concept of a transposition hyperidentity, and extend it to cover arbitrary arityn ≥ 2. We show that theMV hyperidentities with onen-ary operation symbol have no finite basis, but do have a countably infinite basis consisting of these transposition hyperidentities.