Clone Theory

A clone of operations (functions) is a family C of operations on a fixed set such that C contains the projection operations and is closed under superposition. For every algebra A, the clone generated by the basic operations of A, and called the clone of A, captures most structural properties of A. Two algebras with identical clones have the same subalgebras, congruences, automorphisms, etc.

Clones on finite sets are determined by finitary invariant relations; put differently, the clone of a finite algebra A is determined by the subalgebras of finite powers of A. Thus, studying the clones of finite algebras is a way of looking at the finite members of the variety they generate.

The clone of a variety is the clone of the countably generated free algebra in the variety, or equivalently, the clone of any single algebra generating the variety. This concept is related to a number of topics on varieties (e.g., equivalence of varieties, interpretation of varieties into one another, the general theory of Mal'tsev conditions, hyperidentities).


R. Poschel, L. A. Kaluznin, Funktionen- und Relationenalgebren, WEB Deutscher Verlag der Wissenschaften, Berlin, 1979.

Ágnes Szendrei, Clones in Universal Algebra, Seminaire de Mathematiques Superieures, Les Presses de l'Universite de Montreal, Montreal, 1986.

Some Open Problems