Aportes y Recomendaciones

We will now define some concepts which allow us to find certain congruences in any semigroup:

retractable ideals (Definition 5.10) andliftable congruences (Definition 5.11). These construc- tions are new, first appearing in [EMRT18] to help describe some of the congruences onPn and

its submonoids. It will turn out that all non-Rees congruences ofMncan be found using these

two building blocks.

Definition 5.10. LetSbe a finite semigroup, with minimal idealM. An idealI ofS is called

retractableif there exists some homomorphismφ:I→M such that (m)φ=mfor allm∈M; we call φa retraction.

Definition 5.11. LetS be a finite semigroup, with minimal idealM. A congruenceσonM is aliftable congruenceofS if either, and therefore both, of the following equivalent conditions are satisfied:

(i) σ∪∆S is a congruence onS;

(ii) (ax, bx),(xa, xb)∈σfor all pairs (a, b)∈σand elementsx∈S.

To see that the two conditions in the last definition are equivalent, assume we haveS, M and σsuch that (i) is satisfied. Now let (a, b)∈σ andx∈S be arbitrary. Sinceσ∪∆S is a

congruence and (a, b)∈σ∪∆S, we must have (ax, bx)∈σ∪∆S. If (ax, bx)∈∆S thenax=bx,

and since M is an ideal we must have bothax and bxin M; hence (ax, bx) is a reflexive pair and lies in the congruenceσ. If (ax, bx)∈/∆S then (ax, bx)∈σ. Hence, either way, (ax, bx) is

in σ, and by a similar argument, so is (xa, xb), so we have (ii).

Conversely, assume that (ii) holds, let (a, b)∈σ∪∆S, and letx∈S. If (a, b)∈∆S thena=

b, and soax=bxandxa=xb. Otherwise, (a, b)∈σand by (ii) we have (ax, bx),(xa, xb)∈σ. In either case, we have (ax, bx),(xa, xb)∈σ∪∆S, and so we have (i).

In order to use these building blocks to produce new congruences, we first need to establish some results about them. Note that, since Mn is finite, it must have a minimal ideal. More

specifically, the minimal ideal of Mn is given byI0={α∈ Mn: rankα= 0}(see Proposition

5.7). The following lemma will be used at various times throughout this chapter.

Lemma 5.12. Let S be a finite semigroup with minimal ideal M, and letI be an ideal of S. If I is retractable andφ is a retraction fromI toM, then(sxt)φ=s·(x)φ·t for all elements

x∈I and alls, t∈S1_.

Proof. SinceS is a finite semigroup, we know that its minimal idealM is regular, by [How95, Proposition 3.1.4]. Hence any elementm∈M has an elementm0∈M such thatmm0m=m. Since (mm0)m=mwe have a left identity form; and sincem(m0m) =m, we also have a right identity. Letebe a right identity for (x)φ, so that (x)φ·e= (x)φ. Sinceφis a retraction and e, xe∈M, we have

so (x)φ=xe. Now letf be a left identity for (sx)φ; we also have (sx)φ·e=f ·(sx)φ·e = (f)φ·(sx)φ·e = (f sx)φ·e = (f s)φ·(x)φ·e = (f s)φ·(x)φ = (f sx)φ = (f)φ·(sx)φ =f ·(sx)φ = (sx)φ,

which shows thateis a right identity for (sx)φas well as for (x)φ. Hence we have s·(x)φ=s·xe= (sxe)φ= (sx)φ·(e)φ= (sx)φ·e= (sx)φ,

i.e. φrespects left multiplication; a symmetric argument gives (xt)φ= (x)φ·t, i.e.φ respects right multiplication too. Finally we can combine these to give (sxt)φ= (sx)φ·t=s·(x)φ·t, as required.

The previous lemma gives rise to an important corollary which we can use later when we combine retractable ideals with liftable congruences.

Corollary 5.13. Let S be a finite semigroup, with minimal idealM. IfI is a retractable ideal of S, then the retractionφ:I→M is unique.

Proof. Letφ and ψ be retractions from I to M. Letx∈ I, let el be a left identity for (x)φ,

and leterbe a right identity for (x)ψ. By Lemma 5.12, we have

(x)φ=el·(x)φ= (elx)φ=elx= (elx)ψ=el·(x)ψ, so (x)φ=el·(x)ψ. Similarly, (x)ψ= (x)ψ·er= (xer)ψ=xer= (xer)φ= (x)φ·er, so (x)ψ= (x)φ·er. But then (x)φ=el·(x)ψ=el·(x)φ·er= (x)φ·er= (x)ψ, so φ=ψ.

The effect of Corollary 5.13 is that, for a finite semigroup with a regular minimal ideal, we can talk aboutthe retraction of a retractable ideal without any loss of generality. We can now use our two building blocks to produce a new congruence: alifted congruence.

Definition 5.14. LetS be a semigroup with minimal idealM, let I be a retractable ideal of S, and letσbe a liftable congruence ofS. We associate to the pair (I, σ) the relation

ζI,σ=

n

(x, y)∈I×I: (x)φ,(y)φ

whereφis the unique retraction fromI toM. We callζI,σthelifted congruenceof (I, σ).

In order to justify the namelifted congruence, we require the following theorem.

Theorem 5.15. The relationζI,σ in Definition 5.14 is a congruence onS.

Proof. For conciseness, let us refer to ζI,σ as ζ. Let (x, y) be a pair in ζ and let s ∈ S. To

show ζ is a congruence, we must show that (sx, sy) and (xs, ys) both lie inζ. If (x, y)∈∆S,

this is certainly true. Otherwise, we havex, y∈Iand (x)φ,(y)φ

∈σ. SinceIis an ideal, we certainly have sx, sy∈I. Now by Definition 5.11(ii), and by Lemma 5.12, we have

s·(x)φ, s·(y)φ= (sx)φ,(sy)φ∈σ, so (sx, sy)∈ζ. A symmetric argument gives us (xs, ys)∈ζ.

This construction now gives us a usable source of congruences. All that is required is to find some liftable congruences and retractable ideals of a semigroup, and a number of new congruences can be described. It turns out that this is an excellent source of congruences for

Mn, yielding every non-Rees congruence on the semigroup, as we will see later.