Single identities for varieties equivalent to quadruple systems

Abstract If H is a subgroup of the symmetric group S4, then a 3-groupoid (S, f) is called H-permutable if f(xσ(1), xσ(2), xσ(3)) = xσ(4) ↔ f(x1, x2, x3) = x4 for every σ ∈ 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 S4 which contains at least one permutation σ such that σ(4) / 4.