nLab
clone

Clones

Idea

A 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 SS is a clone on SS if it contains all (component) projections S nSS^{n}\to S and is closed under composition (“superposition”).

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

See also the thesis

  • Miles Gould, Coherence for operadic theories, Glasgow 2009 pdf

Last revised on January 26, 2017 at 15:24:36. See the history of this page for a list of all contributions to it.