Singulary cylindric and polyadic equality algebras

Abstract

This paper is a contribution to the structure and representation theory of finitedimensional cylindric and polyadic equality algebras. The main result is that every singulary algebra is representable, where an algebra is singular y if it is generated by elements supported by singletons. In particular, all prime algebras are representable (a prime algebra has no proper subalgebra, and every cylindric or polyadic equality algebra has a prime subalgebra). Since every polyadic algebra can be embedded in a polyadic equality algebra the results also apply to them. Every infinite-dimensional singulary algebra is locally finite and hence is known to be representable, so we shall be concerned just with the finite-dimensional case, where previously very little was known about representation. It may be mentioned that for any dimension greater than one there are nonrepresentable cylindric and polyadic equality algebras(2). For some special kinds of singulary algebras we obtain a direct construction of the representation, which brings out the structure of the algebra clearly. The methods used essentially constitute an algebraic version of Behmann's solution of the decision problem for the singulary predicate calculus with equality(3). Since only finitely many variables are available, the algebraization has some novel features. The essential ideas can be seen in the case of prime algebras, for which no advanced results about cylindric or polyadic algebras are needed. In the general case, however, use is made of the fact that a cylindric algebra is representable if every finitely generated subalgebra of it is. In the last section of the paper we discuss the logical counterpart of the methods and results of the algebraic part of the paper.