congruence relation on an algebraic system

Let (A,O) be an algebraic system. A congruence relationPlanetmathPlanetmath, or simply a congruencePlanetmathPlanetmathPlanetmath on A

  1. 1.

    is an equivalence relationMathworldPlanetmath on A; if (a,b) we write ab(mod), and

  2. 2.

    respects every n-ary operator on A: if ωA is an n-ary operator on A (ωO), and for any ai,biA, i=1,,n, we have

    aibi(mod)   implies   ωA(a1,,an)ωA(b1,,bn)(mod).

For example, A2 and ΔA:={(a,a)aA} are both congruence relations on A. ΔA is called the trivial congruence (on A). A proper congruence relation is one not equal to A2.


  • is a congruence relation on A if and only if is an equivalence relation on A and a subalgebraPlanetmathPlanetmathPlanetmath of the productMathworldPlanetmathPlanetmathPlanetmath ( A×A.

  • The set of congruences of an algebraic system is a complete latticeMathworldPlanetmath. The meet is the usual set intersectionMathworldPlanetmathPlanetmath. The join (of an arbitrary number of congruences) is the join of the underlying equivalence relations ( This join corresponds to the subalgebra (of A×A) generated by the union of the underlying sets of the congruences. The lattice of congruences on A is denoted by Con(A).

  • (restrictionPlanetmathPlanetmath) If is a congruence on A and B is a subalgebra of A, then B defined by (B×B) is a congruence on B. The equivalence of B is obvious. For any n-ary operator ωB inherited from A’s ωA, if aibi(modB), then ωB(a1,,an)=ωA(a1,,an)ωA(b1,,bn)=ωB(b1,,bn)(mod). Since both ωB(a1,,an) and ωB(b1,,bn) are in B, ωB(a1,,an)ωB(b1,,bn)(modB) as well. B is the congruence restricted to B.

  • (extensionPlanetmathPlanetmathPlanetmath) Again, let be a congruence on A and B a subalgebra of A. Define B by {aA(a,b) and bB}. In other words, aB iff ab(mod) for some bB. We assert that B is a subalgebra of A. If ωA is an n-ary operator on A and a1,,anB, then aibi(mod), so ωA(a1,,an)ωA(b1,,bn)(mod). Since ωA(b1,,bn)B, ωA(a1,,an)B. Therefore, B is a subalgebra. Because BB, we call it the extension of B by .

  • Let B be a subset of A×A. The smallest congruence on A such that ab(mod) for all a,bB is called the congruence generated by B. is often written B. When B is a singleton {(a,b)}, then we call B a principal congruence, and denote it by (a,b).

Quotient algebra

Given an algebraic structure (A,O) and a congruence relation on A, we can construct a new O-algebraMathworldPlanetmath (A/,O), as follows: elements of A/ are of the form [a], where aA. We set

[a]=[b] iff ab(mod).

Furthermore, for each n-ary operator ωA on A, define ωA/ by


It is easy to see that ωA/ is a well-defined operator on A/. The O-algebra thus constructed is called the quotient algebra of A over .

Remark. The bracket []:AA/ is in fact an epimorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, with kernel ( ker([])=. This means that every congruence of an algebraic system A is the kernel of some homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath from A. [] is usually written [] to signify its association with .


Title congruence relation on an algebraic system
Canonical name CongruenceRelationOnAnAlgebraicSystem
Date of creation 2013-03-22 16:26:23
Last modified on 2013-03-22 16:26:23
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 33
Author CWoo (3771)
Entry type Definition
Classification msc 08A30
Related topic Congruence3
Related topic Congruence2
Related topic CongruenceInAlgebraicNumberField
Related topic PolynomialCongruence
Related topic QuotientCategory
Related topic CategoryOfAdditiveFractions
Defines congruence
Defines congruence relation
Defines quotient algebra
Defines proper congruence
Defines trivial congruence
Defines non-trivial congruence
Defines congruence restricted to a subalgebra
Defines extension of a subalgebra by a congruence
Defines principal congruence
Defines congruence generated by