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Clone (algebra)

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

  • C contains all the projections πkn: An → A, defined by πkn(x1, …,xn) = xk,
  • 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 h(x1, …,xn) := f(g1(x1, …,xn), …, gm(x1, …,xn)) 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 22κ clones on an infinite set of cardinality κ.

    Abstract clones

    Philip Hall introduced the concept of abstract clone. An abstract clone is different from a concrete clone in that the set A is not given. Formally, an abstract clone comprises

  • a set Cn for each natural number n,
  • elements πk,n in Cn for all k≤n, and
  • a family of functions ∗:Cm × (Cn)m→Cn for all m and n

    • such that

  • c ∗ (π1,n,...,πn,n) = c
  • πk,m ∗ (c1,...,cm) = ck
  • c ∗ (d1 ∗ (e1,...,en),...,dm∗ (e1,...,en)) = (c ∗ (d1,...dm)) ∗ (e1,...,en).

    • Any concrete clone determines an abstract clone in the obvious manner.

    Any algebraic theory determines an abstract clone where Cn is the set of terms in n variables, πk,n are variables, and ∗ is substitution. Two theories determine isomorphic clones if and only if the corresponding categories of algebras are isomorphic. Conversely every abstract clone determines an algebraic theory with an n-ary operation for each element of Cn. This gives a bijective correspondence between abstract clones and algebraic theories.

    Every abstract clone C induces a Lawvere theory in which the morphisms m→n are elements of (Cm)n. This induces a bijective correspondence between Lawvere theories and abstract clones.

    References

    Clone (algebra) Wikipedia
    (Text) CC BY-SA


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