Algoritmo Geométrico de sustracción espectral

We conclude this section with a brief investigation into longest elements of CI-monoids.

We introduce Coxeter groups and make clear the association between the longest element of a Coxeter group and the zero element in its corresponding Coxeter monoid.

An example is then provided of a finite CI-monoid without a unique longest element and whose zero element is not longest.

Definition. Given a monoidM =FX/∼, we say u∈FX is areduced word (for M) if it is a word of minimal length in its ∼-class. Then l(u) =l(u). An elementg∈M is alongest element ofM if it is an element of maximal length inM.

Definition. Suppose M(X, m) is a Coxeter monoid. Then the Coxeter group W(X, m) is the group on generating setX and with relations

(xixj)2m(xi,xj)= 1 for all pairs (xi, xj)∈X×X.

Coxeter groups are well studied in the literature. For on overview, see [1]. Note. The involution relations of the formx2_i = 1 imply that every element of the group may be represented by positive words onX. In other words, a Coxeter group may be considered as a monoid with the same presentation. It makes sense, therefore, to speak of the longest element in a Coxeter group. Lemma 1.5.21. A Coxeter group W has a longest element if and only if it is finite. Moreover, ifW has a longest element it is unique.

Proof. For a proof of this, see for instance [1, p. 36, Prop. 2.2.9, Prop. 2.3.1].

We now make clear the association between the zero element in a Coxeter monoidM(X, m) and the unique longest element in its corresponding Cox- eter groupW(X, m). First we summarize results from [28] in our notation.

Proposition 1.5.22. Suppose M(X, m) is a Coxeter monoid. A word u∈

FX is a reduced forM(X, m)if and only if it is for W(X, m). Furthermore,

reduced words u and v on X are equivalent in M(X, m) if and only if they are in W(X, m). This determines a bijection φ between the elements of

W(X, m) andM(X, m). Proof. See [28, Thm. 1].

Definition. LetM(X, m) be a Coxeter monoid. IfGis a group andH1, H2 are subsets ofGthen let H1H2 :={h1h2 : h1 ∈H1, h2 ∈H2}. Let Γ(X, m) denote the monoid consisting of the subsets of W(X, m) generated by the set {hxi :x ∈ X} of two-element subgroups, with this binary operation of set-wise multiplication inW(X, m).

Lemma 1.5.23. LetM(X, m)be a Coxeter monoid. The mapX →Γ(X, m)

defined byx7→ hxi determines an isomorphism ρ:M(X, m)→Γ(X, m). Proof. See [28, Thm. 1].

The following two lemmas clarify results of [28], and are not claimed as original.

Lemma 1.5.24. Suppose W = W(X, m) is a Coxeter group. Then W is finite if and only if W ∈Γ(X, m).

Proof. For ”only if”, assume W is finite. For sets U and V write U ( V

if U is a proper subset of V. For all subsets W0 ⊆W, either W0 = W or there isx∈X with W0(W0hxi. As W is finite this says that there exists

finite r ≥ 1 and x1, . . . , xr ∈ X such that W0 ( W0hx1i ( W0hx1ihx2i (

. . . ( W0hx1ihx2i. . .hxri = W. As Γ(X, m) is non-empty, this says that W ∈Γ(X, m).

For ”if”, assume W ∈ Γ(X, m). By the definition of Γ(X, m) there exists finite r ≥1 and x1, . . . , xr ∈ X such thatW =hx1i. . .hxri. Then |W|=

|hx1i. . .hxri| ≤ |hx1i|. . .|hxri|= 2r, soW is finite.

Lemma 1.5.25. Letφbe as in Proposition1.5.22andρbe as in Lemma1.5.23. If a Coxeter group W = W(X, m) is finite with longest element w0 then φ(w0) is a zero element of M(X, m). Conversely, if w is a zero element of M(X, m) thenW is finite with longest elementφ−1(w).

Proof. For the first half, note that for each g∈W, we have g∈(ρ◦φ)(g). Also, (ρ◦φ)(g) contains elements of length at mostl(g). Recall thatW has a unique longest element by Lemma 1.5.21. It follows that (ρ◦φ)(w0) is the only element in the image of ρ◦φ containing w0. Now, ρ is surjective

(ρ◦φ)(w0). (?) ThenW is clearly a zero element of Γ(X, m) becausehxiW = W =Whxifor allx∈X. Asρis an isomorphism, it follows by Lemma 1.2.9 thatφ(w0) is a zero element of M(X, m).

For the second part, supposewis a zero element ofM(X, m). Thenρ(w) is a zero element of Γ(X, m) by Lemma 1.2.9. It follows that ρ(w)hxi for all x∈X, soρ(w)g=ρ(w) for allg∈W. In other words,W =ρ(w) andW is finite by Lemma 1.5.24. ThenW has a longest elementw0 by Lemma 1.5.24. Then ρ(w) = W = (ρ◦φ)(w0) by (?). Finally, as ρ is injective, we have w0 =φ−1(w).

Proposition 1.5.22 and Lemma 1.5.25 clarify that a Coxeter monoid M is finite if and only if it has a zero elementw, and moreover thatwis a unique longest element inM.

We have seen already in Lemma 1.3.3 that for n ≥ 3, the CI-monoids Qn are infinite but have zero elements. So Qn is an example of an infinite CI- monoid with a zero element but no longest element, in contrast to the case of Coxeter monoids.

We have shown in Theorem 1.5.1 that every finite CI-monoid has a zero element. Certainly if M is finite then M has a longest element. It might be anticipated that if M is finite then the zero element is always longest. However, the example that will follow Lemma 1.5.26 shows that unlike for finite Coxeter monoids, a finite CI-monoidM does not always have a unique longest element. Moreover, the zero element ofM is not always longest, in contrast to the case of Coxeter monoids.

Lemma 1.5.26. Suppose S is a complete rewriting system for a finitely presented monoid M = FX/ ∼. Suppose also that S respects a shortlex

ordering on X. Then anyS-reduced word on X is reduced.

Proof. Supposeu∈FX isS-reduced but not reduced. Then there isv∈FX with l(v) < l(u) and u ∼ v. Then v is not S-reduced and u 6= v as every

∼-class ofM has a unique reduced element by Theorem 1.4.1. AsS respects a shortlex ordering onX, we must havel(u)≤l(v) a contradiction. So u is reduced.

Example 1.5.27. Let M be the CI-monoid on X ={a, b, c} with CI-graph

a 9 b c . Then M ∼= L3, M has two longest elements, and

neither is zero element.

Proof. Using GAP and KBCA, we obtain the following complete and re- duced rewriting systemS forM with respect to the shortlex ordering≤on FX witha≤b≤c:

a2 →a cbc→bcb cbac→bcba abcbabacba→abcbabacb b2 →b ababa→abab acbaba→acbab ababcbaba→ababcbab c2 →c babab→abab cbabcb→bcbabc bcbabacbab→cbabacbab ca→ac

TheS-reduced words ofM of length at least 9 are summarized below, where wis the zero element. The elementsv1andv2are longest by Lemma 1.5.26

and neither is equal to w.

Length S-reduced word Label

9 ababcbabc w abcbabacb u1 babacbabc u2 babacbabc u3 bcbabacba u4 cbabacbab u5 10 babcbabacb v1 cbabacbabc v2

Table 1.5: TheS-reduced words in L3 of length at least 9.

The relationship between the corresponding elements in M = FX/∼ of Example 1.5.27 is illustrated below in the relevant portion of the double Cayley graph of M. For words u, u0 ∈FX and x ∈X, u x u0 in the

graph if and only if u0 ∼ xu and u x u0 in the graph if and only if u0 ∼ux: u4 v1 u1 u5 u3 w v2 u2 b a c c a c b c a b a _b a b _c a

Figure 1.2: The double Cayley graph ofL3 for elements of length at least 9. Remark. For some finite CI-monoids, a large proportion of the elements have length at least that of the zero element. For example, the finite CI- monoid W4 with CI-graph ◦ ◦ 7 ◦ ◦ is notable in that it has a zero element of length 10 but a longest element of length 18, and 68 of its 304 elements have length at least 10.