Not Finitely Definable Partial Clones on a Finite Set

Abstract

By establishing the relational theory of extendable partial clones on a finite set we describe infinite descending chains of partial clones whose intersection cannot be determined by a finite set of relations (we call them not finitely definable). A special type of such chains introduced in case of clones by I. Rosenberg (1972) as a generalization of two Post clones is investigated.