Editing Congruence relation (section)

== Universal algebra ==
The general notion of a congruence is particularly useful in [[universal algebra]]. An equivalent formulation in this context is the following:{{sfnp|ps=|Bergman|2011|loc=Sect. 1.5 and Exercise 1(a) in Exercise Set 1.26 (Bergman uses the expression ''having the substitution property'' for ''being compatible'')}}

A congruence relation on an algebra ''A'' is a [[subset]] of the [[direct product]] {{nowrap|''A'' × ''A''}} that is both an [[equivalence relation]] on ''A'' and a [[subalgebra]] of {{nowrap|''A'' × ''A''}}.

The [[kernel (universal algebra)|kernel]] of a [[homomorphism]] is always a congruence. Indeed, every congruence arises as a kernel.
For a given congruence ~ on ''A'', the set {{nowrap|''A'' / ~}} of [[equivalence class]]es can be given the structure of an algebra in a natural fashion, the [[quotient (universal algebra)|quotient algebra]].
The function that maps every element of ''A'' to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.

The [[lattice (order)|lattice]] '''Con'''(''A'') of all congruence relations on an algebra ''A'' is [[algebraic lattice|algebraic]].

[[John M. Howie]] described how [[semigroup]] theory illustrates congruence relations in universal algebra:
: In a group a congruence is determined if we know a single congruence class, in particular if we know the normal subgroup which is the class containing the identity. Similarly, in a ring a congruence is determined if we know the ideal which is the congruence class containing the zero. In semigroups there is no such fortunate occurrence, and we are therefore faced with the necessity of studying congruences as such. More than anything else, it is this necessity that gives semigroup theory its characteristic flavour. Semigroups are in fact the first and simplest type of algebra to which the methods of universal algebra must be applied&nbsp;...{{sfnp|ps=|Howie|1975|p=v}}