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4.4. Double centralizing relation 47
We have that:
q(s0d1(σ) · α) · σ = q(s0d1(σ) · α) · q(σ) · s0d0(σ) =
= q(s0d1(σ) · α) · q(σ) · s0d1q(σ)−1 · s0d1q(σ) · s0d0(σ) = (∗).
Now q(σ) · s0d1q(σ)−1 is in K[d1] and hence it commutes with q(s0d1(σ) · α),
while
s0d1q(σ) · s0d0(σ) = s0d1q(σ) · s0d1s0d0(σ) = s0d1(q(σ) · s0d0(σ)) = s0d1(σ).
Then we get
(∗) = q(σ) · s0d1q(σ)−1 · q(s0d1(σ) · α) · s0d1(σ) = q(σ)·s0d1q(σ)−1 ·s0d1(σ)·α=q(σ)·s0d0(σ)·α=σ·α.
4.4 Double centralizing relation

We shall show in the next chapter that any Schreier equivalence relation has a centralizer; we need for that the following precisions (we refer to [13] for a more detailed account). According to the end of Proposition 4.3.3, when we have [R,S] = 0, we necessarily get xSp(xRySz) and p(xRySz)Rz. In set theoretical terms, this means that, with any triple xRySz, we can associate a square of related elements:
x S // p(x,y,z) R // R
y z.
S
More acutely, this says that any connected pair of reflexive relations (R,S) on the monoid X produces the following diagram of reflexive relations in Mon:
R ×X S oo p1 //// S OOR OO
(d0 p0,p) (p,dS1p1) dS0 dS1
(4.4.1)
p0
dR1 //
R oo
// X
dR0
It is called the centralizing double relation associated with the connector and it characterizes the fact that [R,S] = 0. When R is an equivalence relation, the