## 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).

### References

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

• 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 by
• a lattice order,
• a nontrivial equivalence relation,
• a central relation,
• a regular relation.
(The base set is assumed to be finite.)
• Find all maximal TC-clones on a finite set of cardinality greater than 3.
• 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
• a singleton,
• finite,
• countably infinite?
• Is congruence modularity a join prime Mal'cev condition?
• Is the property of having a semilattice operation a join prime Mal'cev condition?
• Is the lattice of clones on a finite set of cardinality >2 simple?