Page 73 - Textos de Matemática Vol. 45

6.3. Schreier internal structures in semirings 65 Definition 6.3.1. An internal reflexive graph in the category of semirings
d0 // X1oos0 //X0
d1
is a Schreier reflexive graph if the split epimorphism (d0 , s0 ) is a Schreier one.
The definitions of Schreier reflexive relation, Schreier internal category and Schreier internal groupoid are analogous.
Thanks to Lemma 6.0.8, the following results are immediate consequences of the analogous ones for monoids.
Example 6.3.2. For every semiring X, the discrete internal equivalence relation: 1X //
Xoo1X //X 1X
is a Schreier internal equivalence relation.
Example 6.3.3. The internal order in SRng given by the usual order between natural numbers:
p0 // ON oo s0 // N,
p1
where
is a Schreier order relation.
ON = {(x, y) ∈ N × N | x ≤ y},
Proposition 6.3.4. Any Schreier reflexive relation is transitive. It is an equiva- lence relation if and only if K[d0] is a ring (i.e. if and only if (K[d0],+,0) is an abelian group).
Proposition 6.3.5. Given a semiring X, the indiscrete equivalence relation ∇X given by:
p1
Corollary 6.3.6. Given any semiring B, the following conditions are equivalent: (a) the semiring B is a ring
(b) any split epimorphism with codomain B is a Schreier split epimorphism.
//
is a Schreier equivalence relation if and only if X is a ring.
p0 X×Xoo s0 //X