Page 38 - Textos de Matemática Vol. 45

30 Chapter 3. Schreier internal relations, categories and groupoids 3.1 Schreier reflexive relations and graphs
Example 3.1.1. For every monoid X, the discrete internal equivalence relation ∆X:
1X // Xoo1X //X
1X
is a homogeneous internal equivalence relation. It is explicitly given by: x1∆Xx2
if and only if x1 = x2.
Example 3.1.2. Example 2.2.7 shows that the internal order in Mon given by
the usual order between natural numbers:
p0 // ON oo s0 // N,
p1
where
is a homogeneous order relation.
ON = {(x, y) ∈ N × N | x ≤ y},
Example 3.1.3. Since Z, with the usual sum, is a group, the internal order in
Mon given by the usual order between integers:
p0 // OZ oo s0 // Z,
p1
where
is a homogeneous order relation (thanks to Proposition 2.2.4).
OZ = {(x, y) ∈ Z × Z | x ≤ y},
The previous example can be obviously generalized to the case of the or-
der relation OG associated with any ordered group G.
Given any Schreier reflexive relation R on a monoid M and any pair xRy,
let us denote by 1Rq(xRy) the value of the Schreier retraction at xRy. Lemma 3.1.4. The characteristic equations of a Schreier retraction become
q(xRy)·x=y and q(xR(t·x))=t.
Proof. The first equality comes from the fact that s0(x) = xRx, hence
xRy = 1Rq(xRy) · xRx = xR(q(xRy) · x).
Concerning the second one, an element of K[d0] is of the form 1Rt, and so we have:
1Rt = 1Rq(1Rt · xRx) = 1Rq(xR(t · x)).