Clone (algebra)

Article summary

In the area of mathematics known as universal algebra, a clone is a set C of finitary operations on a set A such that *C contains all the projections , defined by , *C is closed under (finitary multiple) composition (or "superposition"): if f, g 1 , …, g m are members of C such that f is m-ary, and g j is n-ary for all j, then the n-ary operation is in C. The question whether clones should contain nullary operations or not is not treated uniformly in the literature. The classical approach as evidenced by the standard monographs on clone theory considers clones only containing at-least-unary operations. However, with only minor modifications (related to the empty invariant relation) most of the usual theory can be lifted to clones allowing nullary operations. The more general concept includes all clones without nullary operations as subclones of the clone of all at-least-unary operations and is in accordance with the custom to allow nullary terms and nullary term operations in universal algebra. Typically, publications studying clones as abstract clones, e.g. in the category theoretic setting of Lawvere's algebraic theories, will include nullary operations. 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.…

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