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Algebra Congruence

An algebra congruence on a universal algebra A is an equivalence relation compatible with every defining operation. If an r-ary operation f is part of the universal algebra, compatibility means

a_i∼b_i for every i ==> f(a_1,...,a_r)∼f(b_1,...,b_r).

The operations therefore descend to the set of equivalence classes, giving a quotient universal algebra (Burris and Sankappanavar 1981, pp. 35-37).

For a ring R, each ideal I gives the algebra congruence a∼b iff a-b in I. For example, in Z/4Z, the ideal {0,2} identifies 0 with 2 and 1 with 3. The congruence lattice consists of all algebra congruences ordered by inclusion.

See also

Congruence Lattice, Equivalence Relation, Ideal, Quotient Ring, Universal Algebra

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References

Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. https://math.uwaterloo.ca/~snburris/htdocs/ualg.html.

Cite this as:

Weisstein, Eric W. "Algebra Congruence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraCongruence.html

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