Equational classes of Steiner systems

Abstract

Steiner Triple Systems are in a 1-1-correspondence with the so-called squags (Steiner quasigroups: groupoids satisfying the identities xx=x, xy = yx, x(xy)= y). With the help of this correspondence, many combinatorial properties of Steiner Triple Systems can be described in an algebraic language, and algebraic methods have successfully been applied. Consequently, there have been several attempts to find a similar algebraic theory for Steiner Systems of different types (for definitions, see below). In 1964, Stein published his important paper 'Homogeneous Quasigroups', upon which this paper is based. Almost at the same time, Szamko~owicz in a series of publications tried to classify equational classes of algebras which correspond to Steiner Systems. The present paper considers the 'coordinatization problem' for Steiner Systems from a more universal algebraic point of view. It is easy to find for a given Steiner System a universal algebra, such that the subalgebras are exactly the subsystems of the Steiner System. The problem is now whether there is a 'nice' class of algebras which are all coordinatizing algebras of Steiner Systems such that every Steiner System of a given type has a coordinatizing algebra in that class. We consider a class of algebras to be nice if many algebraic constructions can be made within the class. From this point of view varieties are the nicest classes of algebras and so our problem is whether there are varieties of algebras coordinatizing all Steiner Systems of a given type. Our main result (Theorem 1.2) is rather disappointing: The only varieties having the desired coordinatization properties are essentially those discovered by Stein and Szamkotowicz. In all of these cases we can get these varieties by taking the full idempotent reduct R of some finite field and taking the largest variety having R as a free algebra on two generators. The remainder of the paper is a collection of properties of these varieties. It turns out that they are especially interesting from the algebraic point of view because they have permutable, regular, and normal congruences, the strong amalgamation property, the finite embedding property and thus a solvable word problem, a nice equational base etc.. This is not an 'algebraic theory of Steiner Systems', but hopefully one step towards it. We should remark that three further steps have already been taken in the recent work of Quackenbush, which contains many new ideas and techniques useful for the 'coordinatization' of Steiner Systems and similar structures.