A compelling consequence

By the above mentioned strong amalgamation property, we have that

{et)~\S{ft) n R = t- ^ e S ft n S = {eSf)i{j n S Q S i p n S Q S'i2 n S ii = = Fo.

Therefore there exists u q 6

U

qsuch that the relation t~ ^ezft = u q p holds in K*. Let g, h E

F be such that uqUq^ < g and Uq^uq < h, so that the relation UQp = (gt)~^UQ{ht) = t~^uot holds in K*. Now from t~ ^ezft — t~^UQt and = t~H = 1 it follows that e z f = u q .

This, in turn, implies that 'uq'Uq^ < e and < /; hi other words u q E Uqj. Also,

notice that e z f = uq is a relation between elements of S, and hence it also holds in S^u,tp' So, returning to we have

s = t~^ztf = t~ ^ e zftf = t~'^UQtf E t~^U ejtf,

completing the proof. □

We are now in the position to prove the remaining inclusion Up Ç

U

q

.

Let u E Up. Then there exist e, f E F such that uu~^ < e and u~^u < / . It follows th at t~^utf = up holds in

S*u,(p>

Applying Lemma 5.6.2, we have that there exists u q

E U

qsuch that

up = u q p , and so we obtain that u = u q E

U

q

,

proving that Up —

U

qis indeed finitely

generated. □

We conclude this section by formulating-Proposition 5.6.1 in the special case when E{U) satisfies the maximum condition. Recall that we say that E(H) satisfies the maxi­ mum condition if U has finitely many maximal idempotents F — {fi, f2 ,. . . , fm } and for every idempotent e E E{U) there exists f i E F such that e < fi. Note that in this case F is also a finite J7-dominant subset and Up = U holds, and so we may conclude:

C o rollary 5.6.3. Let S be a finitely presented inverse semigroup and let U, V be inverse subsemigroups of S that are order ideals. Assume that U and V are isomorphic via p : U ^ V . Assume that E{U) satisfies the maximum condition. Then the HNN extension S^u,tp is finitely presented if and only if U is finitely generated.

5.7 A com pelling consequence

Proposition 5.6.1 and its proof have the following intriguing consequence. Assume that S, U, V and p are as in Proposition 5.6.1 and suppose that is finitely presented. Taking any finite j7-dominant set F C E{U), we have

182 C H APTERS. HNN EXTENSIONS where Wo is a finite subset of Wp^Up. But then the converse part of the proof of Proposition 5.6.1 shows how this presentation yields a finite generating set for Up. In other words if Up is finitely generated for some finite j7-dominant subset F then Uq is finitely generated for every finite H-dominant subset G. In fact, we can prove this in full generality removing the requirements that S be finitely presented, that U be an order ideal, and any mention of F.

P ro p o sitio n 5,7.1. Let S be an inverse semigroup and assume that the inverse sub­ semigroup U is finitely U-dominated. Then Up is finitely generated for some finite J - dominant set F if and only if Uq is finitely generated for every finite J-dom inant set G.

The proof is divided into the following two lemmas.

L em m a 5.7.2. Let Up be finitely generated and g be an arbitrary idempotent of U. Then Upu{g} M finitely generated.

Proof. Assume that Up is generated by the finite set A. Let e E E be such that g < j e. Then there exist k,l E U, such that g = kel = kell~^ek~^ = kll~^ell~^k~^, hence there exists v E U such that g = vev~^. From this we have that gv = ve = gve. We

claim that is generated by A U {gve}. Let s E HPu{g}- The following four cases

are to be considered.

(i) The case when s E Up is straightforward.

(ii) Assume that ss“ ^, < g. Then s = 5^55 = •s-ne)n~^. Since eu“ ^S'üe E Hp, there exist a \ , . .. ,ak E A U A“ ^ such that ev~^sve = a i ... a^. From this we have

that 8 = ve ' a \ . . .a^ - ev~^ = gve - u i ... - (gve)~^.

(iii) Assume that < g and s~^s < f for some f E F. Then s — g s f = v{ev~^ • sf).

Since ev~^sf E Up, there exist &%,..., E A U A“ ^ such that ev^^sf = a i ... a/..

Fi'om this we have that s — ve - a i ... ak = gve • a i ... aj^.

(iv) The case when s'~^s < g and ss~^ < f for some f E F can be proved similarly as

(iii).

□

L em m a 5.7.3. Assume that Up is finitely generated and that for some g E F , G = E \{^}

Vf.*

5.8. EXAM PLES 183

Proof. Let Hp be generated by the finite subset A which we may assume, without loss of generality, is closed under taking inverses. There exists e E F \ { g } for which

g 6, and hence g = v e v ~ ^ for some v E U. Let A' denote the set of elements of A

for which a a ~ ^ < g and for which there is no other j E F \ { g } such that a a ~ ^ < j\ also

let A" = Notice that if a € A' then a = g a , while if a G A", then a = a g . Let

Ê = A \ (A' U A"). We show that Uq is generated by

B = ê c { a i v e : G A \ A'} U G A \ A^'} U G A}.

Let s E Uq Ç Up. Then s — aiag ... a^ for some a i , . . . , a^ G A. If a^ G A', then substitute a* by gap, if a* G A", then substitute a% by a*g; and if a^ G A' n A", then substitute a, by gaig. Consider now all subwords w of the form ai-iaigai+ig... gam-igamUm+i^ These subwords can be written in terms of B, since

w = ai-i • aive • ev~^aive • ev~^ai^i... • ev~^am-iue • • am+i>

It follows that s can be written in terms of B, hence Uq is generated by B. □

5.8 E xam ples

Here we provide four examples of HNN extension of inverse semigroups.

E xam ple 1. Let B be the bicyclic monoid, considered as the set N x N with binary operation {m,n){p,q) = (m — n + m ax(n,p),g — p + max(n,p)). Now B is presented as an inverse monoid by B = Inv(aja = a^a~^), where a = (0,1). Let H = B and V = {{m ,n)\m ,n > 0} with p : U V the shift map (p,q) (p + l,g + 1), so that ap = a “ ^a^. It is clear that U — B is H-dominated by the identity element 1 = (0,0), and that U[iy = B is finitely generated. Hence the HNN extension B^s,ip is finitely presented. It is easy to check that

Inv(a,f|a = — aa~^,t~^t = a~^a,t~^at = a“ ^a^) gives a presentation.

E xam ple 2. W ith B as before, suppose now that H = B(B) = {(m ,m )|m G N} with p : (m, m) t-» (m + 1, m + 1). It is clear that H is finitely H-dominated by the identity element 1 = (0,0) of H, but H^jj. = U is not finitely generated. Hence we may conclude that the HNN extension B=i^u,(p is finitely generated but not finitely presented.

184 C H APTERS. HNN EXTENSIONS E xam ple 3. In this example we construct a finitely presented HNN extension S*u,(p, with U not finitely generated. Let S — Inv(a, e|aa~^ = l,ae = 0, = e) considered as a semigroup with 0. It can be easily seen that

G N} U {0}

is a set of normal forms for S. Consider the inverse subsemigroup U ~ {a~''ea^i,j G N} U {0}

of S , together with the identity isomorphism l: U -^ U . We observe that U is isomorphic to the infinite aperiodic Brandt semigroup, and hence U is H-dominated by any of its idempotents. Let F = {a~^ea^}. Then Up — {a~^ea^,0} is finite, so certainly finitely generated, and it follows that the HNN extension is finitely presented. However, the associated inverse subsemigroup U is not finitely generated.

E xam ple 4. Our final example shows that the maximum condition on E{U) is essential in Corollary 5.6.3. We give an example of a finitely presented HNN extension of a finitely presented inverse semigroup 5, where the associated subsemigroups contain finitely many maximal idempotents, but do not satisfy the maximum condition and are not finitely gen­ erated. Consider the semidirect product S — Yqo X G, introduced in Example 4.10.5. We verified that S is finitely presented as an inverse semigroup. In particular, it is generated by the set {(eo, 1), (l,p ), and if we correspond a to (1,^), b to (1,5“ ^) and c to (eo, 1), then we have that

S — Inv(a, 6, c\ab = ba = 1,(? — c, cbc = be, cac = ca). Then it can be easily seen that

(i) {a* : 7 G N} U {F : z G N} U {a6} U {x'^cy^ : x, y e {a, 6} i, j G N} is a set of normal forms for S,

(ii) E{T) is an infinite chain with an identity element adjoined on top:

< b^ca^ < bca < c < ach < < ... < 1,

(iii) the H-class H i of 1 is isomorphic to the infinite cyclic group,

(iv) K = T \ H i is a bisimple aperiodic inverse subsemigroup of S, which is not finitely generated.

It follows from (ii) and (iv) that, for any e G E{S) \ {!}, the subsemigroup Ue = {u e U: uu~^,u~^u < e}

5.8. EXAM PLES 185

is finitely generated. To be more accurate, if e = (e^, 1), then Uq is generated by the finite set 1),

Take two copies RH) and of

S

and let

T

be their 0-direct union. It is straightfor­ ward that T is finitely presented. Let / G and let

U

be the 0-direct union of

K

with

Uf.

Then

U

has one maximal idempotent / , and is not finitely generated since the subsemigroup

K

contains an infinite ascending chain of idempotents. Moreover since

K

is bisimple, for any e G

E{K),

the idempotents

e

and / are H-maximal in

U

and

U{ej} — {u e U: uu~^,u~^u < e

or

uu~^,u~^u

< /}

is isomorphic to the 0-direct union of Hg and

Uf,

and hence is finitely generated. It follows by Proposition 5.6.1, that the HNN extension

H

= S^u,l, where l:

U

—7

U

is the identity isomorphism, is finitely presented.

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In document Automatic S acts and inverse semigroup presentations (Page 194-200)