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Birkhoff's Theorem


Let A and B be two algebras over the same signature Sigma, with carriers A and B, respectively (cf. universal algebra). B is a subalgebra of A if B subset= A and every function of B is the restriction of the respective function of A on B.

The (direct) product of algebras A and B is an algebra whose carrier is the Cartesian product of A and B and such that for every f in Sigma and all x_1,...,x_n in A and all y_1,...,y_n in B,

 f(<x_1,y_1>,...,<x_n,y_n>)=<f(x_1,...,x_n),f(y_1,...,y_n)>.

A nonempty class K of algebras over the same signature is called a variety if it is closed under subalgebras, homomorphic images, and Cartesian products over arbitrary families of structures belonging to the class.

A class of algebras is said to satisfy the identity s=t if this identity holds in every algebra from this class. Let E be a set of identities over signature Sigma. A class K of algebras over Sigma is called an equational class if it is the class of algebras satisfying all identities from E. In this case, K is said to be axiomatized by E.

Birkhoff's theorem states that K is an equational class iff it is a variety.


See also

Birkhoff's Ergodic Theorem, Poincaré-Birkhoff-Witt Theorem, Universal Algebra, Variety

This entry contributed by Alex Sakharov (author's link)

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References

Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.

Referenced on Wolfram|Alpha

Birkhoff's Theorem

Cite this as:

Sakharov, Alex. "Birkhoff's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BirkhoffsTheorem.html

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