Clones

# Clones

## Idea

An (abstract) clone is a cartesian multicategory with one object, in the same way that an operad is a symmetric multicategory with one object. Thus, a clone could equivalently be called a cartesian operad. Equivalently, a clone is a presentation of a single-sorted algebraic theory in terms of algebraic operations, equivalent to a Lawvere theory but organized slightly differently.

## Definition

A set of algebraic operations on a fixed set $S$ is a (concrete) clone on $S$ if it contains all (component) projections $S^{n}\to S$ and is closed under composition (“superposition”).

An abstract clone consists of an abstract set of “$n$-ary operations” for every $n$ together with projection and composition operations. For now, see wikipedia. This is the notion that’s equivalent to a cartesian operad or a Lawvere theory.

## References

• Ágnes Szendrei, Clones in universal algebra, Séminaire de mathématiques supérieures 99, Les presses de l’université de Montreal, 1986. — 166 p.

A rather general framework is discussed in

• Zhaohua Luo, Clone theory, its syntax and semantics, applications to universal algebra, lambda calculus and algebraic logic, arxiv/0810.3162
• Dietlinde Lau, Function algebras on finite sets: Basic course on many-valued logic and clone theory, Springer Monographs in Mathematics

A common generalization of a clone and of an operad is proposed, using a new notion of a verbal category, in

• S. Tronin, Abstract clones and operads, Siberian Mathematical Journal 43, No.4, 746–755, 2002 link

Another unification of clones and operads is via the formalism in

• Pierre-Louis Curien, Operads, clones, and distributive laws, arxiv/1205.3050