Variety (universal algebra)


Variety (universal algebra)

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 which 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.

Contents

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 equations, asserting equality between terms. A model satisfies these equations if they are true in the model for any 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 much harder.

Examples

The class of all semigroups forms a variety of algebras of signature (2). A sufficient defining equation is the associative law:

x(yz) = (xy)z.

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:

x(yz) = (xy)z
1x = x1 = x
xx − 1 = x − 1x = 1.

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 xy = yx, 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.

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 semigroups and hence in formal language theory. Eilenberg's theorem, often referred to as the variety theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.

Category theory

If A is a finitary algebraic category, then the forgetful functor

U:A\to\bold{Set}

is monadic. Even more, it is strictly monadic, in that the comparison functor

K:A\to \bold{Set}^{\mathbb{T}}

is an isomorphism (and not just an equivalence).[1] Here, \bold{Set}^{\mathbb{T}} is the Eilenberg–Moore category on \bold{Set}. In general, one says a category is an algebraic category if it is monadic over \bold{Set}. This is a more general notion than "finitary algebraic category" (the notion of "variety" used in universal algebra) because it admits such categories as CABA (complete atomic Boolean algebras) and CSLat (complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity. The algebraic category of sigma algebras also has infinitary operations, but their arity is countable whence its signature is small (forms a set).

See also

Notes

  1. ^ Saunders Mac Lane, Categories for the Working Mathematician, Springer. (See p. 152)

References

Two monographs available free online:


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Universal algebra — (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ( models ) of algebraic structures.For instance, rather than take particular groups as the object of study, in universal… …   Wikipedia

  • Variety — may refer to: *Variety (botany), a rank in botany below that of species. *Variety (cybernetics), the number of possible states of a system or of an element of the system. *Variety (linguistics), a concept that includes for instance dialects,… …   Wikipedia

  • Universal property — In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. These properties are called universal properties …   Wikipedia

  • Algebra — This article is about the branch of mathematics. For other uses, see Algebra (disambiguation). Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from… …   Wikipedia

  • Algebra of Communicating Processes — The Algebra of Communicating Processes (ACP) is an algebraic approach to reasoning about concurrent systems. It is a member of the family of mathematical theories of concurrency known as process algebras or process calculi. ACP was initially… …   Wikipedia

  • Universal algebraic geometry — In Universal algebraic geometry, algebraic geometry is generalized from the geometry of rings to geometry of arbitrary varieties of algebras, so that every variety of algebras has its own algebraic geometry. Note that the two terms algebraic… …   Wikipedia

  • Algebraic variety — This article is about algebraic varieties. For the term a variety of algebras , and an explanation of the difference between a variety of algebras and an algebraic variety, see variety (universal algebra). The twisted cubic is a projective… …   Wikipedia

  • Interior algebra — In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and… …   Wikipedia

  • List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia

  • Mathematical variety — In mathematics the meaning of variety can be in algebraic geometry, an algebraic variety, which may be affine, projective or abstract or in universal algebra, a variety, a set of structures satisfying some further given set of equations on their… …   Wikipedia


We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.