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. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories. 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. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories. 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 a collection of algebras, defined by a set of equations that must be identically satisfied in each algebra, while an algebraic variety is a collection of elements from a single algebra, defined by a set of equations that these elements must satisfy. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common. The term 'variety of algebras' refers to algebras in the general sense of universal algebra; there is also a more specific sense of algebra, namely as algebra over a field, i.e. a vector space equipped with a multiplication. A signature (in this context) is a set, whose elements are called operations, each of which is assigned a natural number (0, 1, 2,...) called its arity. Given a signature σ {displaystyle sigma } and a set V {displaystyle V} , whose elements are called variables, a word is a finite planar rooted tree in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation o {displaystyle o} has as many branches away from the root as the arity of o {displaystyle o} . An equational law is a pair of such words; we write the axiom consisting of the words v {displaystyle v} and w {displaystyle w} as v = w {displaystyle v=w} . A theory is a signature, a set of variables and a set of equational laws. Any theory gives a variety of algebras as follows. Given a theory T {displaystyle T} , an algebra of T {displaystyle T} consists of a set A {displaystyle A} together with, for each operation o {displaystyle o} of T {displaystyle T} with arity n {displaystyle n} , a function o A : A n → A {displaystyle o_{A}colon A^{n} o A} such that for each axiom v = w {displaystyle v=w} and each assignment of elements of A {displaystyle A} to the variables in that axiom, the equation holds that is given by applying the operations to the elements of A {displaystyle A} as indicated by the trees defining v {displaystyle v} and w {displaystyle w} . We call the class of algebras of a given theory T {displaystyle T} a variety of algebras. However, ultimately more important than this class of algebras is the category of algebras and homomorphisms between them. Given two algebras of a theory T {displaystyle T} , say A {displaystyle A} and B {displaystyle B} , a homomorphism is a function f : A → B {displaystyle fcolon A o B} such that for every operation o {displaystyle o} of arity n {displaystyle n} . Any theory gives a category where the objects are algebras of that theory and the morphisms are homomorphisms. The class of all semigroups forms a variety of algebras of signature (2), meaning that a semigroup has a single binary operation. A sufficient defining equation is the associative law: