Distributive groupoids and the finite basis property

Abstract

Recently, the finite basis property of varieties of algebras has often been investigated. Special attention has been paid to varieties of quasigroups. Evans ( J. Algebra 31 (1974), 508–513) proved that every finitely generated commutative Moufang loop is finitely based. This result was used by Mendelsohn and Padmanabhan ( J. Algebra 49 (1977), 154–161) to show that every finitely generated commutative distributive groupoid satisfying, for some n ⩾ 2, the identity x( x( x(··· ( xy)))) = y with the variable x repeating n times (such groupoids are necessarily quasigroups) is finitely based. Here we present two more general results. Namely, we show that each finitely generated distributive (even trimedial) quasigroup as well as each finitely generated commutative distributive groupoid is finitely based.