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.

Read more about Variety (universal Algebra):  Birkhoff's Theorem, Examples, Pseudovariety of Finite Algebras, Category Theory

Famous quotes containing the word variety:

    The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.
    Calvin Coolidge (1872–1933)