|
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature that is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras is usually called a finitary algebraic category. A covariety is the class of all coalgebraic structures of a given signature. A variety of algebras should not be confused with an algebraic variety. Intuitively, a variety of algebras is an equationally defined ''collection of algebras'', while an algebraic variety is an equationally defined ''collection of elements from a single algebra''. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common. ==Birkhoff's theorem== Garrett Birkhoff proved equivalent the two definitions of variety given above, a result of fundamental importance to universal algebra and known as Birkhoff's theorem or as the HSP theorem. H, S, and P stand, respectively, for the closure operations of homomorphism, subalgebra, and product''. An equational class for some signature Σ is the collection of all models, in the sense of model theory, that satisfy some set ''E'' of universally quantified ''equations'', asserting equality between terms. A model ''satisfies'' these equations if they are true in the model for every valuation of the variables. The equations in ''E'' are then said to be identities of the model. Examples of such identities are the commutative law, characterizing commutative algebras, and the absorption law, characterizing lattices. It is simple to see that the class of algebras satisfying some set of equations will be closed under the HSP operations. Proving the converse —classes of algebras closed under the HSP operations must be equational— is harder. ==Examples== The class of all semigroups forms a variety of algebras of signature (2). A sufficient defining equation is the associative law: :: It satisfies the HSP closure requirement, since any homomorphic image, any subset closed under multiplication and any direct product of semigroups is also a semigroup. The class of groups forms a class of algebras of signature (2,1,0), the three operations being respectively ''multiplication'', ''inversion'' and ''identity''. Any subset of a group closed under multiplication, under inversion and under identity (i.e. containing the identity) forms a subgroup. Likewise, the collection of groups is closed under homomorphic image and under direct product. Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities. In fact, the familiar axioms of associativity, inverse and identity form one suitable set of identities: : : : A subvariety of a variety ''V'' is a subclass of ''V'' that has the same signature as ''V'' and is itself a variety. Notice that although every group becomes a semigroup when the identity as a constant is omitted (and/or the inverse operation is omitted), the class of groups does ''not'' form a subvariety of the variety of semigroups because the signatures are different. On the other hand the class of abelian groups is a subvariety of the variety of groups because it consists of those groups satisfying with no change of signature. Viewing a variety ''V'' and its homomorphisms as a category, a subclass ''U'' of ''V'' that is itself a variety is a subvariety of ''V'' implies that ''U'' is a full subcategory of ''V'', meaning that for any objects ''a'', ''b'' in ''U'', the homomorphisms from ''a'' to ''b'' in ''U'' are exactly those from ''a'' to ''b'' in ''V''. On the other hand there is a sense in which Boolean algebras and Boolean rings can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely; in this sort of situation the homomorphisms between corresponding structures are the same. 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「Variety (universal algebra)」の詳細全文を読む スポンサード リンク
|