Page 53 - Textos de Matemática Vol. 45

4.3. Centrality for Schreier relations 45 is a (necessarily unique) monoid homomorphism p: R ×B S → B such that
p(xRxSy) = y and p(xRySy) = x. We denote this situation by [R, S] = 0. Example 4.3.2. Given the order ON on N, we have [ON,ON] = 0 introducing
p(x ≤ y ≤ z) = z − y + x.
Suppose S is a Schreier reflexive relation. Given any ySz, we shall denote
by 1Sq(ySz) the value of the associated Schreier retraction S → K[d0].
Proposition 4.3.3. The reflexive relation R and the Schreier reflexive relation S centralize each other in Mon if and only if, for any 1St ∈ K[d0] and any xRy, we have q(yS(y · t)) · x = x · t. In this circumstance we have p(xRySz) = q(ySz)·x. When [R, S] = 0, we have necessarily xSp(xRySz) and p(xRySz)Rz.
Proof. The condition of the existence of p and its definition is a straighforward consequence of Proposition 2.4.3. We check p(xRySy) = q(ySy) · x = 1 · x = x and p(xRxSz) = q(xSz) · x = z.
We know that, in a Mal’tsev category, two reflexive relations (i.e. two equivalence relations) R and S on X centralize each other as soon as R ∩ S = ∆X. In Mon, we have as well:
Proposition 4.3.4. The equivalence relation R and the Schreier equivalence re- lation S on the monoid X centralize each other in Mon as soon as R∩S = ∆X.
Proof. Suppose that 1St and xRy. From the first point we get xS(x·t); for the same reason we have xS(q(yS(y · t)) · x) and since S is an equivalence relation we get (x·t)S(q(yS(y·t))·x). On the other hand, from xRy, we get (x·t)R(y·t) and (q(yS(y · t)) · x)R(q(yS(y · t)) · y) = (q(yS(y · t)) · x)R(y · t); since R is an equivalence relation we get (x · t)R(q(yS(y · t)) · x). Now since R ∩ S = ∆X , we have q(yS(y · t)) · x = x · t and [R, S] = 0.
Given a relation R, we denote by Rop the relation obtained from R by twisting d0 and d1. In other terms, xRopy if and only if yRx.
Proposition 4.3.5. Suppose both Rop and S are Schreier reflexive relations on a monoid X. Then the reflexive relations R and S centralize each other if and only if the submonoids dR0 kdR1 : K[dR1 ] X and dS1 kdS0 : K[dS0 ] X commute in X. Suppose that both R and S are Schreier equivalence relations, then R and S centralize each other if and only if the equivalence classes 1R and 1S commute in X.
Proof. It is a straighforward consequence of Theorem 2.4.6.
Proposition 4.3.6. Let (A,B,f,s) be a Schreier split epimorphism such that R[f] is a Schreier equivalence relation. We have [R[f],R[f]] = 0 if and only if its kernel K[f] is an abelian group.