Rational equivalence of algebras, its clone generalizations, and clone categoricity

Abstract

The clones of functions on a set A are related to an arbitrary universal algebra \(\mathfrak{A} = \left\langle {A;\sigma } \right\rangle \) in various natural ways. The simplest and minimal of them is the clone \(Tr(\mathfrak{A})\) of termal functions of the algebra \(\mathfrak{A}\), that is, the closure of the collection of signature functions of these algebras with respect to the operator of superposition. The coincidence of similar clones (up to conjugation by the bijections of the underlying sets of the algebras) generates various equivalence relations on universal algebras, the first of which is the relation of rational equivalence introduced by Mal’tsev. This article deals with clone equivalences of this kind between algebras of arbitrary signatures.