Clone (algebra)
Encyclopedia
In universal algebra
, a clone is a set C of operation
s on a set A such that
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 22κ clones on an infinite set of cardinality κ.
's PhD thesis introduced the concept of Lawvere theory
, which is the categorical
equivalent of a clone.
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, a clone is a set C of operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
s on a set A such that
- C contains all the projections , defined by ,
- C is closed under (finitary multiple) composition (or "superposition"): if f, g1, …, gm are members of C such that f is m-ary, and gj is n-ary for every j, then the n-ary operation is in C.
Given an algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
in a signature
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...
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
Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....
clones on a finite set of size at least three, and 22κ clones on an infinite set of cardinality κ.
Category theory
William LawvereWilliam Lawvere
Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...
's PhD thesis introduced the concept of Lawvere theory
Lawvere theory
In category theory, a Lawvere theory is a category which can be considered a categorical counterpart of the notion of an equational theory.-Definition:...
, which is the categorical
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
equivalent of a clone.