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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 , defined by , *''C'' is closed under (finitary multiple) composition (or "superposition"〔Denecke, Klaus. ''Menger algebras and clones of terms,'' East-West Journal of Mathematics 5 2 (2003),179-193.〕): if ''f'', ''g''1, …, ''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 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''.〔P. M. Cohn. Universal algebra. D Reidel, 2nd edition, 1981. Ch III.〕 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 π''k'',''n'' in ''Cn'' for all ''k''≤''n'', and *a family of functions ∗:''Cm'' × (''Cn'')''m''→''Cn'' for all ''m'' and ''n'' such that * c ∗ (π''1'',''n'',...,π''n'',''n'') = c * π''k'',''m'' ∗ (c''1'',...,c''m'') = ck * c ∗ (d''1'' ∗ (e''1'',...,e''n''),...,d''m''∗ (e''1'',...,e''n'')) = (c ∗ (d''1'',...d''m'')) ∗ (e''1'',...,e''n''). 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, π''k'',''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Clone (algebra)」の詳細全文を読む スポンサード リンク
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