Let A be an algebraic category.
Let X be an object. Denote by X2 the product X X. If U is a subset of X2 we denote by p, q: U --> X the maps induced by the projections X2 --> X.
Remark 4.1. Recall that an equavlence
relation on a set X is a subset U of X2
satisfying the following conditions: (1) (a, a)
U for any a in X.
Definition 4.2. (a) An equavelence relation
on X is called a congruence
if it is a closed subset of X2.
Proof. If t: X --> T is a morphism then X T X is a closed subset of X2 by (3.6), which is also an equavelence relation on X by (4.1.c).
Proof. Since U = X X/U X by (4.1.a), if q is a morphism then U is an effective congruence. Conversely, assume U = X T X for a morphism t: X --> T, then X/U = t(X) by (4.1) and X --> t(X) is a morphism by (2.5).
Proof. This follows from the fact that the classes of equavelnce relations and closed subsets are closed under intersections.
Proof. For any equivalence relation u, v: U -->X
let q: X --> X/U be the quotient map. Let
--> G be the generic extensiion of q. Then
gq: X -->
G is a morphism. Let C(U) = X G
X. Then C(U) is an effective congruence. We show that C(U)
is the smallest effective congruence containing
U. Suppose V
is another effective congruence on X containing U. The quotient
map p: X --> X/V factors through q: X --> X/U
by a map t: X/U --> X/V. Then by the property of the generic
extention g there is a unique morphism s: G --> X/V
such that t = sg. Thus p = tq =sgq, which shows that V