Clone (algebra)

In universal algebra, a clone is a set C of operations on a set A such that

• C contains all the projections π_kⁿ: Aⁿ → A, defined by π_kⁿ(x₁, …,x_n) = x_k,
• C is closed under (finitary multiple) composition (or "superposition"^[1]): if f, g₁, …, g_m are members of C such that f is m-ary, and g_j is n-ary for every j, then the n-ary operation h(x₁, …,x_n) := f(g₁(x₁, …,x_n), …, g_m(x₁, …,x_n)) is in C.

Given an algebra in a signature σ, the set of operations on its carrier definable by a σ-term (the term functions) is a clone. Conversely, every clone can be realized as the clone of term functions in a suitable algebra.

If A and B are algebras with the same carrier such that every basic function of A is a term function in B and vice versa, then A and B have the same clone. For this reason, modern universal algebra often treats clones as a representation of algebras which abstracts from their signature.

There is only one clone on the one-element set. The lattice of clones on a two-element set is countable, and has been completely described by Emil Post (see Post's lattice). Clones on larger sets do not admit a simple classification; there are continuum clones on a finite set of size at least three, and 2^2κ clones on an infinite set of cardinality κ.

Category theory

William Lawvere's PhD thesis introduced the concept of Lawvere theory, which is the categorical equivalent of a clone.

References

1. ^ Denecke, Klaus. Menger algebras and clones of terms, East-West Journal of Mathematics 5 2 (2003),179-193.

• Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor, Algebras, Lattices, Varieties, Vol. 1, Wadsworth & Brooks/Cole, Monterey, CA, 1987.
• F. William Lawvere: Functorial semantics of algebraic theories, Columbia University, 1963. Available online at Reprints in Theory and Applications of Categories

This algebra-related article is a stub. You can help Wikipedia by expanding it.v ·

This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Algebra stubs Mathematical logic stubs Universal algebra Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Khalid Latif Space Chimps (video game) Look at other dictionaries: Clone — Contents 1 Biological 2 Computing 3 Entertainment 3.1 Music …   Wikipedia List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia Universal algebra — (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ( models ) of algebraic structures.For instance, rather than take particular groups as the object of study, in universal… …   Wikipedia List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia Post's lattice — In logic and universal algebra, Post s lattice denotes the lattice of all clones on a two element set {0, 1}, ordered by inclusion. It is named for Emil Post, who published a complete description of the lattice in 1941 [E. L. Post, The two valued …   Wikipedia Boolean algebras canonically defined — Boolean algebras have been formally defined variously as a kind of lattice and as a kind of ring. This article presents them more neutrally but equally formally as simply the models of the equational theory of two values, and observes the… …   Wikipedia Timeline of computing hardware 2400 BC–1949 — History of computing Hardware before 1960 Hardware 1960s to present Hardware in Soviet Bloc countries Artificial intelligence Computer science Operating systems Programming languages …   Wikipedia Calvin (Calvin and Hobbes) — Infobox comic strip character character= Calvin caption= Calvin in a yelling mood. comic= Calvin and Hobbes creator= Bill Watterson first= November 18, 1985Calvin is a fictional character in the comic strip Calvin and Hobbes by Bill Watterson. He …   Wikipedia Functional completeness — In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression.[1][2] A well known complete set of… …   Wikipedia Complement (set theory) — In set theory, a complement of a set A refers to things not in (that is, things outside of), A. The relative complement of A with respect to a set B, is the set of elements in B but not in A. When all sets under consideration are considered to be …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”