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
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.
Clones in Universal Algebra, Seminaire
de Mathematiques Superieures, Les Presses de l'Universite de Montreal,
Some Open Problems
Given a finite set of operations on a finite set, is it decidable whether
the clone they generate contains a near unanimity operation?
For which finite bounded posets is the (maximal) clone of all monotone
operations finitely generated?
Describe all maximal proper subclones of each maximal clone determined
(The base set is assumed to be finite.)
a lattice order,
a nontrivial equivalence relation,
a central relation,
a regular relation.
Find all maximal TC-clones on a finite set of cardinality greater than
Find a Rosenberg-type characterization for quasiprimal algebras.
Determine all minimal clones (on a finite set).
Is it possible to make a strong link between minimal semiprojection clones
and minimal conservative semiprojection clones via absorption identities?
If C is a minimal clone and n is the least arity of a nontrivial
operation of C, then is it true that
C has only finitely
many n-ary operations?
For which transformation monoids M on a finite set is the interval
of all clones with unary part M
Is congruence modularity a join prime Mal'cev condition?
Is the property of having a semilattice operation a join prime Mal'cev
Is the lattice of clones on a finite set of cardinality >2 simple?