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A Course in Universal Algebra

Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems

The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) ≡ z (modΘ(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z ∈ A} is a transversal of the set of equivalence classes of Θ(a, b); (3)p(a, b, z) andp(a′,b′,z) define the same transversal wheneverΘ(a,b)=Θ(a′,b′); (4)Θ(p(x, y, 1), 1)= Θ(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.

On the structure of varieties with equationally definable principal congruences III

Steiner quadruple systems - a survey

We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability results, and constructions of model companions. The task is considerably simplified by introducing a new definition of sheaf constructions over Boolean spaces. We start the paper by giving a new formulation in §1 of what sheaf constructions over Boolean spaces are-perhaps an appropriate name for our construction would be 'Boolean product'. In §2 we show that this construction is closely related to the formation of reduced products, in particular the universal Horn class generated by a class is the class of structures which can be embedded into the Boolean products of the given class. A specialized Boolean product operator F is indeed remarkably similar to reduced products, for in §3 it is shown that V preserves a sentence iff it is equivalent to a Horn sentence; and in §4 we take another look at Comer's version of the Feferman and Vaught Theorem (for sheaves) and show that it is the most general of those currently in use. Several model-theoretic results proved by Waszkiewicz and Weglorz for reduced limit powers have natural extensions to Boolean products as one also sees in §4. In §5 we take a construction introduced by Arens and Kaplansky, which we call a filtered Boolean power, and show how to extend a translation introduced by Ershov to such structures-this translation is a considerable simplification of the Feferman and Vaught approach of §4. Then in §6 we prove the result which motivated us to look at the elementary properties of sheaf constructions, namely that countable Boolean products of finitely many finite structures with a finite language have a decidable theory. This generalizes Comer's results for monadic algebras and (xm = x)-rings, and Ershov's results for bounded Boolean powers. The next section contains technical results needed for the study of model Received by the editors September 17, 1976 and, in revised form, February 8, 1977. AMS (MOS) subject classifications (1970). Primary 02H13, 02G05; Secondary 02G10, 02G15, 02G20, 08A20.

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Sheaf constructions and their elementary properties

Publisher Summary The chapter describes the techniques used to co-ordinatize Steiner Systems by algebras. The examples in this chapter establishes a correspondence between all Steiner Systems of a fixed type (1, k) and a class of algebras' defined by a set of identities. Every such algebra co-ordinatizes a Steiner System of the respective type and each (t, k)-Steiner System is the underlying system of some (sometimes many) algebras in the class. The co-ordinatization is unique only for sloops, squags, and SQS-skeins (for squags and SQS-skeins it is even functorial). Except for near boolean algebras, every co-ordinatizating algebra can be obtained from a Steiner System.

Co-Ordinatizing Steiner Systems

Produkte von Kongruenzklassengeometrien universeller Algebren

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.

Heinrich Werner

Papers

Co-Ordinatizing Steiner Systems

Produkte von Kongruenzklassengeometrien universeller Algebren

Equational classes of Steiner Systems

Equational compactness in quasi-primal varieties

Diagonal-products and affine completeness