Variety (universal algebra)

In the mathematical subject of 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 the equivalence of the two definitions of a 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, satisfied by commutative algebras, and the absorption law, satisfied by 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:

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^{-1}x=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.

Pseudovariety of finite algebras

Since varieties are closed under arbitrary direct products, all non-trivial varieties contain infinite algebras. Attempts have been made to develop a finitary analogue of the theory of varieties.

A pseudovariety is usually defined to be a class of algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. Not every author assumes that all algebras on a pseudovariety are finite; if this is the case, one sometimes talks of a variety of finite algebras. For pseudovarieties, 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.^[1]

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\colon A\to {\mathbf {Set}}

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

K\colon A\to {\mathbf {Set}}^{\mathbb {T} }

is an isomorphism (and not just an equivalence).^[2] Here, ${\mathbf {Set}}^{\mathbb {T} }$ is the Eilenberg–Moore category on ${\mathbf {Set}}$ . In general, one says a category is an algebraic category if it is monadic over ${\mathbf {Set}}$ . 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).

Notes

↑ E.g. Banaschewski, B. (1983), "The Birkhoff Theorem for varieties of finite algebras", Algebra Universalis, Volume 17(1): 360-368, DOI 10.1007/BF01194543
↑ Saunders Mac Lane (1971), Categories for the Working Mathematician, Springer. (See p. 152.)

References

Birkhoff, G. (1935), "On the structure of abstract algebras", Proceedings of the Cambridge Philosophical Society, 31: 433–454 .

Two monographs available free online:

Stanley N. Burris and H.P. Sankappanavar (1981), A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2. [Proof of Birkhoff's Theorem is in II§11.]
Peter Jipsen and Henry Rose (1992), Varieties of Lattices, Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8.

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