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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 The equivalence of the two definitions given above is of fundamental importance in universal algebra. It was proved by Garrett Birkhoff , and is generally known as Birkhoff's theorem, or as the '''HSP theorem''' (H, S and P standing respectively for the closure operations of ''homomorphism'', ''subalgebra'' and ''product''). Formally, suppose we fix a Signature Σ. An equational class for Σ is the set of all models, in the sense of Model Theory for example, that satisfy ''equations'' in a given set ''E''. Those equations are statements of equality between terms. A model is said to ''satisfy'' them if they are true in the model for any valuation of the variables. Such equations are then said to be Identities of the model. An example is the Commutative Law , which is an identity of commutative algebras; another example is the Absorption Law , which is an identity of Boolean Algebra . It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations, so the burden of Birkhoff's theorem is the converse: classes of algebras that satisfy those conditions must be equational. EXAMPLES The class of all Semigroup s 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 (ie. 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: : : : Notice that although every group is a semigroup, the class of groups does not form a subvariety of the variety of semigroups. This is because not every sub'''semi'''group of a group is a group. The class of Abelian Group s, considered again with signature (2,1,0), also has the HSP closure properties. It forms a subvariety of the variety of groups, and can be defined equationally by the three group axioms above together with the commutativity law: : VARIETY OF FINITE ALGEBRAS Since varieties are closed under arbitrary cartesian products, all non-trivial varieties contain infinite algebras. It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case. With this in mind, attempts have been made to develop a finitary analogue of the theory of varieties. A variety of finite algebras, sometimes called a '''pseudovariety''', is usually defined to be a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. There is no general finitary counterpart to Birkhoff's theorem, but in many cases the introduction of a more complex notion of equations allows similar results to be derived. Pseudovarieties are of particular importance in the study of finite Semigroup s and hence in Formal Language Theory . Eilenberg's Theorem , often referred to as the ''variety theorem'' describes a natural correspondence between varieties of Regular Language s and pseudovarieties of finite semigroups. CATEGORY THEORY If ''A'' is a finitary algebraic category, then the Forgetful Functor : is Monadic . Even more, it is ''strictly monadic'', in that the Comparison Functor : is an isomorphism (and not just an equivalence).Saunders Mac Lane,''Categories for the Working Mathematician'', Springer. ''(See p. 152)'' Here, is the Eilenberg-Moore Category on . In general, one says a category is an algebraic category if it is monadic over . REFERENCES Two monographs available free online:
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