4. CongruencesLet Let X
X. If U is a subset of X we denote by ^{2}p,
q: U --> X the maps induced by the projections X.
^{2}
--> X
a, a)
U for any a in X.
(2) If ( a, b)U
then (b, a)U.
(3) If ( a, b), (b, c)U,
then (a, c)U.
(a) If U is an equavelence relation on X, we denote by
X/U
the quotient set of X under U, and q: X --> X/U
the canonical surjective map. Then U = X .
_{X/U}
X(b) A subset U of X is an equivalence relation
iff ^{2}U = X for a map _{T}
Xt: X --> T (we may assume t is surjective).
(c) Any intersection of equivalence relations is an equivalence.
(b) A subset U of X is called an ^{2}effective
congruence on X if there is a (surjective) morphism t:
X --> T such that U = X .
_{T}
X
X by (3.6), which is also
an equavelence relation on ^{2}X by (4.1.c).
X .
q is a morphism then U is an effective
congruence. Conversely, assume U = X for a morphism _{T}
Xt: X --> T, then X/U = t(X)
by (4.1) and X --> t(X) is a morphism by (2.5).
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
contains C(U).
Next consider the intersection U of a set {U}
of effective congruences on _{i}X. We have C(U)C(U)_{i}
= U for any _{i}U. It follows that _{i}C(U)
is contained in the intersection of {U}, which is _{i}U.
This shows that U is an effective congruence.
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