Variety (algebra)

See also: Variety

In universal algebra, a variety is a nonempty class K of algebraic structures of the same signature such as

  • all subalgebra of an element of K is in K
  • any image by Homomorphisme of an element of K is in K
  • all Produit direct elements of K is also in K

A theorem of Birkhoff (1935) states that the varieties are the équationnelles classes , i.e. the classes of algebras which satisfy a whole of identities (called équationnelle axiomatization of the class).

For example following equations

(X * there) * Z = X * (there * Z)
X * E = X
E * X = X
(for all X, there, Z ) define the variety of the Monoïde S.

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