Clone classification of dually discriminator algebras with finite support

Abstract

ABSTRACT. Dually discriminator algebras are considered up to clones generated by the algebra operations. In terms of binary relations, all clones of the operators on a finite set that contain the Pixley dual discriminator are efficiently described. As a consequence, a similar clone classification of quasi-primal algebras with finite support is determined.KEY WORDS: dual discriminator, dually discriminator algebra, clone, closure of a relation, conjunction of relations, projection of relations, identification.For any set E, by a dual discriminator [1] on E we mean the function d(x,y,z)= { x forx=y, z for x r y.Following [2], we say that an algebra (E ; F) with the support E and the set of operations F is dually discriminator if d is a term operation in the algebra (E; F). Algebras (E; F) and (E ; G) are said to be equivalent [3] if the clones [F] and [G] of operations generated by the sets F and G coincide.Dually discriminator algebras play a noticeable role in the theory of universal algebras and multivalued logic, in particular, in the problems of classification of completeness [1,2,[4][5][6][7][8][9]. For instance, nearly a "half' of all homogeneous algebras [10] (including those with infinite supports) are equivalent to appropriate dually discriminator algebras of finite type [2,[5][6][7].In [1], dually discriminator algebras with finite support are characterized via projectively rectangle subalgebras in Cartesian squares of algebras. However, the description in [1] is not effective even for algebras of finite type, because it requires the verification of certain conditions for the set of all finitary operations. Furthermore, as can readily be seen, there is a continuum of pairwise not isomorphic dually discriminator algebras with finite non-one-element support. In this connection, the problem of efficient classification of such algebras arises.Another reason that stimulates the classification of dually discriminator algebras is that any quasiprimal algebra [11] is a dually discriminator algebra. Thus, we can simultaneously obtain the classification of quasiprimal algebras with finite support as well. Finally, a clone [d] is an atom in the lattice of clones of operations on the corresponding set (see, e.g., [2]), and the principal filter (of the lattice) determined by the clone [d] contains many clones that are of importance in applications.In the present paper, we perform the classification of the dually discriminator algebras with finite support on the basis of the equivalence relation introduced below (the clone classification). In fact, for any finite set E, we describe (Theorem 1) all clones of the operations on E that include the operation d. The description is efficient and is based on the Galois correspondence for Post algebras [12]. Note that, for a chosen finite set E, the finiteness of the clone classification follows from the results of [4]. In Theorem 2 we give a similar classification for the quasiprimal algebras with finite support.For any k, k _> 2, for a standard k-element set we consider t...

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