Single identities for varieties equivalent to quadruple systems

If H is a subgroup of the symmetric group $4, then a 3-groupoid (S, f ) is called H-permutable if f ( x ~ ( 1 ) , X a ( 2 ) , X ~ r ( 3 ) ) = X~(4) ~ f ( x l , x2 ,x3 ) = x4 for every cr C H. Some classes of generalized idempotent H-permutable 3-groupoids are equivalent to Steiner, Mendelsohn and other quadruple systems. We prove that the variety of generalized idempotent H-permutable 3-groupoids can be defined by a single identity for every subgroup H of $4 which contains at least one permutation ~r such that or(4) ¢ 4. I . I n t r o d u c t i o n It is well known that some classes of Steiner, Mendelsohn and other triple and quadruple systems are equivalent to certain algebras. Some of these algebras can be defined by a system of identities [3]. A question which was considered by several authors is when the varieties of such algebras can be defined by a single identity. In [6] it was proved that the variety o f quasigroups equivalent to Mendelsohn triple systems is one-based and in [2] that the same is true for the so-called squags and sloops which are equivalent to Steiner triple systems. That two subvarieties of squags are also one-based was proved in [1]. Each of the varieties o f algebras equivalent to Steiner, Mendelsohn and tetrahedral quadruple systems can also be defined by a single identity, as it was proved in [10,9]. All these algebras have some common properties, which will be described later, and we shall show that the quoted results are special cases o f a more general theorem and they can be extended to include a much wider class of algebras. * Corresponding author. E-mail: [email protected]. 0012-365X/98/$19.00 Copyright (~) 1998 Elsevier Science B.V. All rights reserved PH S00 12-365X(97)00060-5 278 Z. StojakoviL W.A. DudeklDiscrete Mathematics 183 (1998) 277-284