Horn classes of predicate systems and varieties of partial algebras

Abstract

We propound an approach through which techniques of the theory of quasivarieties of predicate systems are brought to bear on partial algebras. For every partial algebra A, two predicate representations are treated. The first is the graph of A whose basic operations are graphs of the basic operations on A. The second representation results from the graph of A by adding domains of the operations on A to its basic relations. Studying partial algebras from various perspectives makes it necessary to deal with different equality semantics. Here we present a general definition of semantics that stretches over such instances as weak semantics, Evans’ semantics. Kleene semantics, and strong semantics. On a set of all semantics, the preorder is induced in increasing “force,” and it is proved that certain of the properties of varieties of partial algebras in a given semantics are individuated by the position it takes in that set. We argue that every variety of partial algebras, in any semantics, is in correspondence with a Horn class of predicate systems which admits a generation operator and is closed under direct limits and retracts. For such classes we prove analogs of the Birkhoff theorem on subdirect decompositions and of the Taylor theorem on residual smallness. Therefore, these are also applicable to varieties of partial algebras in arbitrary semantics.