Universal Algebra (original) (raw)

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Universal algebra studies common properties of all algebraic structures, including groups, rings, fields,lattices, etc.

A universal algebra is a pair A=(A,(f_i^A)_(i in I)), where A and I are sets and for each i in I, f_i^A is an operation on A. The algebra A is finitary if each of its operations is finitary.

A set of function symbols (or operations) of degree n>=0 is called a signature (or type). Let Sigma be a signature. An algebra A is defined by a domain S (which is called its carrier or universe) and a mapping that relates a function f:S^n->S to each n-place function symbol from Sigma.

Let A and B be two algebras over the same signature Sigma, and their carriers are A and B, respectively. A mapping phi:A->B is called a homomorphism from A to B if for every f in Sigma and all x_1,...,x_n in A,

phi(f(x_1,...,x_n))=f(phi(x_1),...,phi(x_n)).

If a homomorphism phi is surjective, then it is called epimorphism. If phi is an epimorphism, thenB is called a homomorphic image of A. If the homomorphism phi is a bijection, then it is called an isomorphism. On the class of all algebras, define a relation ∼ by A∼B if and only if there is an isomorphism from A onto B. Then the relation ∼ is an equivalence relation. Its equivalence classes are called isomorphism classes, and are typically proper classes.

A homomorphism from A to B is often denoted as phi:A->B. A homomorphism phi:A->A is called an endomorphism. An isomorphism phi:A->A is called an automorphism. The notions of homomorphism, isomorphism, endomorphism, etc., are generalizations of the respective notions in groups, rings, and other algebraic theories.

Identities (or equalities) in algebra A over signature Sigma have the form

s=t,

where s and t are terms built up from variables using function symbols from Sigma.

An identity s=t is said to hold in an algebra A if it is true for all possible values of variables in the identity, i.e., for all possible ways of replacing the variables by elements of the carrier. The algebra A is then said to satisfy the identity s=t.

See also

Algebra

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

Portions of this entry contributed by Matt Insall (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.Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.Wolfram, S.A New Kind of Science. Champaign, IL: Wolfram Media, p. 1171, 2002.

Referenced on Wolfram|Alpha

Universal Algebra

Cite this as:

Insall, Matt and Sakharov, Alex. "Universal Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.https://mathworld.wolfram.com/UniversalAlgebra.html

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