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- Clone_(algebra) abstract "In universal algebra, a clone is a set C of finitary operations on a set A such thatC 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.".
- Clone_(algebra) wikiPageExternalLink index.html.
- Clone_(algebra) wikiPageID "15535840".
- Clone_(algebra) wikiPageLength "4286".
- Clone_(algebra) wikiPageOutDegree "15".
- Clone_(algebra) wikiPageRevisionID "690203359".
- Clone_(algebra) wikiPageWikiLink Cardinality_of_the_continuum.
- Clone_(algebra) wikiPageWikiLink Category:Universal_algebra.
- Clone_(algebra) wikiPageWikiLink Emil_Leon_Post.
- Clone_(algebra) wikiPageWikiLink First-order_logic.
- Clone_(algebra) wikiPageWikiLink Function_composition.
- Clone_(algebra) wikiPageWikiLink Lawvere_theory.
- Clone_(algebra) wikiPageWikiLink Operation_(mathematics).
- Clone_(algebra) wikiPageWikiLink Philip_Hall.
- Clone_(algebra) wikiPageWikiLink Posts_lattice.
- Clone_(algebra) wikiPageWikiLink Projection_(set_theory).
- Clone_(algebra) wikiPageWikiLink Set_(mathematics).
- Clone_(algebra) wikiPageWikiLink Signature_(logic).
- Clone_(algebra) wikiPageWikiLink Term_algebra.
- Clone_(algebra) wikiPageWikiLink Universal_algebra.
- Clone_(algebra) wikiPageWikiLinkText "Clone (algebra)".
- Clone_(algebra) wikiPageWikiLinkText "Clone".
- Clone_(algebra) wikiPageWikiLinkText "clone".
- Clone_(algebra) wikiPageWikiLinkText "clones".
- Clone_(algebra) wikiPageWikiLinkText "finitary multiple composition".
- Clone_(algebra) subject Category:Universal_algebra.
- Clone_(algebra) hypernym C.
- Clone_(algebra) type SoccerClubSeason.
- Clone_(algebra) comment "In universal algebra, a clone is a set C of finitary operations on a set A such thatC 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. ".
- Clone_(algebra) label "Clone (algebra)".
- Clone_(algebra) sameAs Q838048.
- Clone_(algebra) sameAs Clone_(mathématiques).
- Clone_(algebra) sameAs m.03mdhph.
- Clone_(algebra) sameAs Клон_(алгебра).
- Clone_(algebra) sameAs Q838048.
- Clone_(algebra) sameAs 克隆_(数学).
- Clone_(algebra) wasDerivedFrom Clone_(algebra)?oldid=690203359.
- Clone_(algebra) isPrimaryTopicOf Clone_(algebra).