The Mobius Inverse Monoid

JOURNAL OF ALGEBRA 200, 428᎐438Ž. 1998 ARTICLE NO. JA977242

The Mobius¨ Inverse Monoid

Mark V. Lawson

School of Mathematics, Uni¨ersity of Wales, Bangor, Gwynedd LL57 1UT, United Kingdom

Communicated by Peter M. Neumann

Received March 20, 1997

We show that the Mobius¨ transformations generate an F-inverse monoid whose maximum group image is the Mobius¨ group. We describe the monoid in terms of McAlister triples. ᮊ 1998 Academic Press

1. INTRODUCTION

Mobius¨ transformations can be handled in two ways. The naive way views them as partial functions defined on the complex plane, whereas the sophisticated way views them as functions defined on the extended complex plane. In the first part of this paper, I shall view Mobius¨ transformations from the naive standpoint. Formally, a Mobius¨ transformation is a partial function ␣ of the complex plane, ރ, having the following form: ␣Ž.z s Žaz q b .Žr cz q d .where a, b,c, d are complex numbers and ad y bc / 0. Of course, there are many ways to write the function ␣ because multiplying a, b, c, and d by a non-zero complex number results in a different form but the same function. This must always be borne in mind in the sequel. Mobius¨ transformations are partial functions rather than functions because of their denominators; if the coefficient c is non-zero then there is one value of z satisfying cz q d s 0. Thus the transformations are of two types: the functions in which c s 0, and the partial functions in which c / 0. Of course, the partial transformations are ‘‘only just’’ partial; the domain of the function omits the point ydrc and the image omits the point arc. The condition ad y bc / 0 ensures that in both cases the transformations are injective. Thus Mobius¨ transformations are either bijections or partial bijections of ރ. The inverse transformation is also a 1 Mobius¨ transformation, namely ␣y Ž.z s Ždz y b .Žrycz q a ..

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0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. THE MOBIUS¨ INVERSE MONOID 429

Mobius¨ transformations can be ‘‘multiplied.’’ If ␣Ž.z s Žaz q b .Žr cz q d.Ž.Ž.Ž.and ␤ z s Az q B r Cz q D are Mobius¨ transformations then their product, which we shall denote by ␣᭪␤, is defined to be ŽŽaA q bC . z q Ž..ŽŽ.Ž..aB q bD r cA q dC z q cB q dD . The result is certainly a Mobius¨ transformation, as can easily be checked, consequently ᭪ is a well-defined binary operation on the set of Mobius¨ transformations. With respect to this operation the Mobius¨¨ transformations form a group, called the Mobius group. Many textbooks make misleading comments concerning this definition. For example, inwx 5, p. 210 we find the statement

Moebius transformations form a group under composition of mappings.

Inwx 7, p. 121 , we find the statement

If one thinks of the Mobius¨ transformation as associated withwx a matrix, then it is easily checked that the effect of composing two Mobius¨ transformations is to multiply the corresponding matrices. The composite of two Mobius¨ transformations is therefore another Mobius¨ transformation.

To see why these comments are misleading, consider the following two specific transformations: ␣Ž.z s Ž2 z q 1 .Žr z y 1 . and ␤ Ž.z s Ž2 z q 1.Žrzq1 . . As partial functions ␣: ރ _ Ä41 ª ރ _ Ä42 and ␤: ރ _yÄ1 4ª ރ_Ä42 . We compute their composition, ␣ ( ␤, as partial functions. Ob- 1 serve that domŽ.␣ l im Ž␤ .s ރ _ Ä41, 2 and so domŽ.␣ ( ␤ s ␤yŽރ _ Ä41, 2 .s ރ _ Ä40, y1 and imŽ.Ž␣ ( ␤ s ␣ ރ _ Ä41, 2 .s ރ _ Ä42, 5 . We see that the partial function composition of ␣ and ␤ is not a Mobius¨ transformation, because two points are omitted from the domain of the resulting function. Thus the composition of two Mobius¨ transformations is not the same as their product. Evidently, there is more going on here than the mathematics is making explicit. The first aim of this paper is to show how the product ᭪ of Mobius¨ transformations can be derived in a natural way from their composition. To do this, we consider the inverse monoid generated by the Mobius¨ transformations regarded as partial functions of ރ. In the second part of the paper, we analyse the structure of this inverse monoid.

2. THE MOBIUS¨ GROUP VIA INVERSE MONOIDS

We shall assume the reader is familiar with elementary inverse semigroup theory as described inwx 2 . At the end of the paper we shall use McAlister triples which are discussed inwx 6 . If S is an inverse semigroup then ESŽ.will denote the set of idempotents of S. 430 MARK V. LAWSON

Recall that the symmetric in¨erse monoid IŽ. X on the set X is the semigroup of all partial bijections of the set X under the operation of composition of functions. The natural partial order in IXŽ.is just the usual inclusion of partial functions. The idempotents of IXŽ.are the identity functions defined on the subsets of X. If 1AB and 1 are identity functions defined on the subsets A and B, respectively, then their composite, 1AB(1 , is the identity function defined on the subset A l B wx2 . The following will be a useful inverse subsemigroup of IXŽ..

LEMMA 1. Let X be an infinite set. The set of bijections between the co-finite subsets of X forms an in¨erse subsemigroup of IŽ. X . Proof. If X _ A and X _ B are co-finite subsets of X then Ž.X _ A l Ž.X_BsX_ ŽAjB .is co-finite. It is now straightforward to prove that the bijections between the co-finite subsets form an inverse subsemigroup. Remark. We shall refer to the elements of the above inverse semigroup as ‘‘co-finiteŽ. partial bijections .’’ Since Mobius¨ transformations are partial bijections of ރ we can regard them as elements of the symmetric inverse monoid IŽ.ރ . The finite compositions of Mobius¨ transformations and their inverses generate an inverse submonoid of IŽ.ރ , which we call the Mobius¨ in¨erse monoid, denoted M.

LEMMA 2. The idempotents of M consist of the identity functions defined on the co-finite subsets of ރ. Proof. Each element of M is a finite composition of Mobius¨ transformations each of which is co-finite. Thus every element of M is co-finite. Hence every idempotent of M is co-finite. We now prove that every co-finite idempotent of IŽ.ރ is an element of M. Let w be any complex number. Every subset of the form ރ _ Ä4w can be obtained as the domain of some Mobius¨ transformation in which w sydrc. Finally, the identity defined on the subset ރ _ Ä4w1,...wn can be obtained by composing the identities defined on the subsets ރ _ Ä4w1 ,...,ރ_ Ä4wn . The proof of the following is immediate from the definitions.

LEMMA 3. ␣ ( ␤ : ␣᭪␤. The following property of Mobius¨ transformations is fundamental. See wx1 for a proof. THEOREM 4. A Mobius¨ transformation which fixes three or more points is the identity. Hence two Mobius¨ transformations which agree on three or more points must be equal. THE MOBIUS¨ INVERSE MONOID 431

The above result has the following important consequence for the structure of M.

LEMMA 5. If ␣ g M and ␣ : ␤, ␥ where ␤ and ␥ are Mobius¨ transformations, then ␤ s ␥. Proof. The functions ␤ and ␥ agree on the set domŽ.␣ . By Lemma 2, this set is co-finite and so, in particular, contains at least three elements. The result now follows from Theorem 4.

DEFINITION. A semigroup is said to be E-unitary if whenever e is an idempotent and e F s then s is an idempotent. PROPOSITION 6. The Mobius¨ transformations generate an E-unitary in- ¨erse submonoid M of IŽ.ރ . The maximal elements of M are the Mobius¨ transformations, and e¨ery element of M lies beneath a unique maximal element. Proof. By Lemma 3, the composition of two Mobius¨ transformations is beneath a Mobius¨ transformation. Thus the composition of any finite number or Mobius¨¨ transformations is beneath a Mobius transformation, and so every element of M is beneath a Mobius¨ transformation. The Mobius¨ transformations form the maximal elements of M with respect to the natural partial order, for if ␣ : ␤ where ␣ is a Mobius¨ transformation and ␤ an arbitrary element of M, then ␤ : ␥ for some Mobius¨ transformation ␥. Thus ␣ : ␤. But then by Lemma 5, ␣ s ␤. We have thus proved that the maximum elements in the poset Ž.M, : are precisely the Mobius¨ transformations. That every element of M is beneath a unique maximum element follows by Lemma 5. Thus M is E-unitary.

DEFINITION. The minimum group congruence ␴ is defined on an inverse semigroup S by Ž.s, t g ␴ iff there exists an element u such that u F s and u F t. The following is well known, but we include a proof for the sake of completeness.

LEMMA 7. Let S be an in¨erse semigroup. If a, b F c thenŽ. a, b g ␴ . Proof. From the properties of the natural partial order, we have that 1 1 1 1 1 a s aay c and b s bby c. Put d s aay bby c. Then d s aay b F b and 1 1 1 dsbby aay c s bby a F a. Hence Ž.a, b g ␴ . LEMMA 8. Two elements of M are ␴-related iff they are bounded abo¨eby the same maximal element. Proof. Let ␣ and ␤ be two elements of M which are ␴-related. By assumption there is an element ␥ such that ␥ : ␣ and ␥ : ␤. Now ␣ and 432 MARK V. LAWSON

␤ lie beneath maximal elements ␣ X and ␤ X of M which are Mobius¨ transformations. Thus ␥ lies beneath ␣ X and ␤ X. But then by Lemma 5, X X X X ␣ s ␤ , and so ␣, ␤ : ␣ s ␤ . On the other hand, by Lemma 7, two elements which are bounded above must be ␴-related. It is now clear that each ␴-class contains a unique Mobius¨ transformation which is the maximum element of its ␴-class.

DEFINITION. An inverse monoid is called F-in¨erse if each ␴-class contains a maximum element.

THEOREM 9. The Mobius¨ in¨erse monoid is F-in¨erse, and Mr␴ is isomorphic to the Mobius¨ group. Proof. By Lemma 8, M is F-inverse. Consider now the group Mr␴ . The elements of this group are in bijective correspondence with the maximum elements of the ␴-classes and so with the Mobius¨ transformations. Let ␴␣Ž.and ␴␤ Ž .be two ␴-classes containing the maximum elements ␣ and ␤. Their product in Mr␴ will be a ␴-class ␴␥Ž. containing the maximum element ␥. Clearly, ␣ ( ␤ : ␥. But ␣ ( ␤ : ␣᭪␤ by Lemma 3, ␣᭪␤ is a Mobius¨ transformation, and ␣ ( ␤ is beneath a unique Mobius¨ transformation by Proposition 6. Thus ␥ s ␣᭪␤.

3. THE MOBIUS¨ INVERSE MONOID

We shall now investigate the inverse semigroup structure of M.Itis tempting to think this must be rather simple, perhaps that it is a semidirect product of a semilattice by a group. We shall show first that this is not the case.

DEFINITION. Let S be an inverse monoid with group of units USŽ.. Then S is said to be factorisable if for each s g S there exists g g USŽ. such that s F g. The following is stated as an exercise inwx 6, p. 346 . An explicit proof may be found inwx 3 .

THEOREM 1. An in¨erse semigroup is isomorphic to a semidirect product of a semilattice by a group if, and only if, SisE-unitary and for each a g S 1 1 and idempotent e g S there exists b g S such that bby s e and ay bisan idempotent. We shall use the following later.

COROLLARY 2. An in¨erse monoid which is E-unitary and factorisable is isomorphic to a semidirect product of a semilattice by a group. THE MOBIUS¨ INVERSE MONOID 433

Proof. We show that the conditions of Theorem 1 hold. Let a g S and ean idempotent. Since S is factorisable there exists a g g USŽ.such that 1 1 1 a F g. Thus a s gay a. Put b s eg. Clearly, bby s e, and ay b s 1 1 Ž.gay ay eg s Ž.Ž.aЈagЈeg , an idempotent. The above result is discussed in more detail inwx 3 .

PROPOSITION 3. M is not a semidirect product of a semilattice by a group.

Proof. Let ␣ be any Mobius¨ transformation which omits a point w1 from its domain. Let ␫ be the identity transformation. By Theorem 1, if M were a semidirect product, we would be able to find an element ␤ such 1 1 that ␤ ( ␤y s ␫ and ␣y ( ␤ is an idempotent. Now ␤ must be beneath a Mobius¨ transformation, ␥ say, by Proposition 2.6. But then ␤ s 1 1 ␤ ( ␤y (␥ s ␥. Thus ␤ is a Mobius¨ transformation. Since ␤ ( ␤y s ␫,it 1 is easy to see that ␤ must be a bijection on ރ. Now ␣y ( ␤ is an 1 1 idempotent and so ␣y : ␤y . Hence ␣ : ␤. But ␣ and ␤ are both Mobius¨ transformations and so must be equal by Lemma 2.5. However, this contradicts the fact that ␤ is a bijection and ␣ omits a point.

Although M is not a semidirect product of a semilattice by a group, we shall show that M can be nicely embedded in such a semidirect product, and so obtain a description of M in terms of McAlister tripleswx 6 . To do this, we allow ourselves to use the sophisticated approach to Mobius¨ transformations. We adjoint a new symbol ϱ to the set ރ and denote the resulting set by ރU. This new symbol is required to satisfy the usual arithmetic properties of infinity: z q ϱ s ϱ s ϱ q z, for all z g ރ; zϱ s ϱ s ϱz for all zgރ_Ä40; zrϱs0 and zr0 s ϱ for all z g ރ _ Ä40 , and ϱϱ s ϱ. The inverse semigroup IŽ.ރ will be regarded as an inverse subsemigroup U of IŽރ .Ž. Each Mobius¨ transformation ␣ z.s Žaz q b.rŽcz q d.can be extended to a bijection ␣ U of ރ, called an extended Mobius¨ transformation, U U in the following way: if c / 0, then ␣ Ž.ydrc s ϱ and ␣ Ž.ϱ s arc and U elsewhere agrees with ␣, whereas if c s 0 then ␣ Ž.ϱ s ϱ and elsewhere agrees with ␣.

LEMMA 4.

U Ž.i For e¨ery Mobius¨ transformation ␣, we ha¨e that ␣ s 1ރރ( ␣ (1. UUU Ž.ii If ␣ and ␤ are Mobius¨ transformations then Ž.␣᭪␤ s ␣ ( ␤ . Proof.

Ž.i Let ␣ be the Mobius¨ transformation ␣Ž.Ž.Ž.z s az q b r cz q d . U If c s 0 then the result is clear since ␣ Ž.ϱ s ϱ. Suppose that c / 0. 434 MARK V. LAWSON

U Then a simple calculation shows that imŽ 1ރރ( ␣ (1 . s ރ _ Ä4arc , and U domŽ 1ރރ( ␣ (1 .s ރ _yÄ4drc. Ž.ii This is a straightforward calculation. The extended Mobius¨ transformations form a subgroup H of the group of units of IŽރU ., which is isomorphic to the Mobius¨ group. LEMMA 5. Eëry element of M is beneath a unique element of H. Proof. By Proposition 2.6, every element of M is beneath a unique Mobius¨ transformation. Thus every element of M is beneath an element of H. Now suppose that ␥ is an element of M and lies beneath the extended Mobius¨ transformations ␣ U and ␤ U. Then ␣ U and ␤ U agree on the set domŽ.␥ . Thus ␣ and ␤ agree on this set. But then by Lemma 2.5, UU we must have that ␣ s ␤. Hence ␣ s ␤ . Let M X be the inverse subsemigroup of IŽރU . generated by M and H. LEMMA 6. Let M be an inërse subsemigroup of an inërse monoid I, and let H be a subgroup of the group of units of I. Suppose that each element of M X lies beneath an element of H. Let M be the inërse subsemigroup of I X generated by M and H. Then M is a factorisable inërse monoid with group of units H. Proof. Each element of MX is a product of elements from M and H. Thus each element of MX lies beneath an element of H in the natural partial order. Clearly the identity of H is the identity of MX, thus H is a subgroup of the group of units of MX. Thus MX is factorisable. On the other hand, any invertible element of MX would have to lie beneath an element of H. But the order on the group of invertible elements is trivial. Thus H is the group of units of MX. It follows from the above result that M X is factorisable with group of units H. The elements of M and H are co-finite. Thus M X is an inverse subsemigroup of the inverse semigroup of co-finite elements of IŽރU .. X X LEMMA 7. M is E-unitary, and eëry element of M lies beneath a unique element of H. Proof. Since every idempotent is defined on a co-finite subset every idempotent fixes more than three points of ރ. But any element above an idempotent is also beneath an extended Mobius¨ transformation ␣ U. Thus ␣ will also fix more than three points, and so will be the identity. Thus ␣ U will be the identity. Thus M X is E-unitary. Now suppose that ␥ is an element of M X lying beneath two extended Mobius¨ transformations ␣ U and ␤ U. Since domŽ.␥ is a co-finite set, it is clear that ␣ and ␤ agree on UU at least three points. Hence ␣ s ␤ and so ␣ s ␤ . THE MOBIUS¨ INVERSE MONOID 435

Notation. We shall extend the notation introduced earlier by also X denoting the unique element of H which is above ␣ g M in the natural partial order by ␣ U. U LEMMA 8. Let w / ϱ be an element of ރ . Then there is an extended U U Mobius¨ transformation ␣ , such that the image of the element ␣ (1ރ is U U ރ_Ä4w and the domain of ␣ (1ރ is ރ. In particular, the identity defined U on ރ _ Ä4wisD-related to 1.ރ UU Proof. Consider the element ␣ of H given by ␣ Ž.z s wzr Žz q 1if . U w/0, and by ␣*Ž.z s 1r Žz q 1if . ws0. Observe that ␣ Ž.y1 s ϱ U and ␣ Ž.ϱ s w. A simple calculation shows that the image of the element UU U ␣(1isރރރ_Ä4w. Clearly, the domain of ␣ (1isރ. The remainder of the lemma is standard inverse semigroup theory.

THEOREM 9. Ž.i MX is isomorphic to some semidirect product of a semilattice by a group. Ž.ii The idempotents of M Xare the co-finite idempotents of IŽރU.. Ž.iii M X is isomorphic to a semidirect product of the semilattice of co-finite idempotents of IŽރU . by the group H. Proof. Ž.i We have shown that M X is both E-unitary and factorisable. Thus by Corollary 2, it is isomorphic to a semidirect product of a semilattice by a group. Ž.ii We now show that every co-finite idempotent of IŽރU.belongs X X to M . First, M contains the idempotent 1ރ since this is the identity of M. UU Let w / ϱ be an element of ރ . Then by Lemma 8, ރ _ Ä4w is the image U X of the element ␣ (1ރ , which belongs to M . Thus every idempotent which is defined on the set ރU with one element removed belongs to M X. The result now follows by taking all finite compositions of these idempotents. Ž.iii Denote the semilattice of idempotents of M X by E. Define a 1 1 function H = E ª E by Ž.Ž.Ž.g, e ¬ ge ge ys gegy. This is clearly a well-defined action of H on E. Now put S s E = H, and define a product 1 on S by Ž.Ž.Že, gf,hsengfgy, gh.. Thus S is an inverse semigroup and a semidirect product of E by H. We prove that M X is isomorphic to S. Define a function ␾: MЈ ª S by

1 U ␾␣Ž.s Ž␣(␣y,␣ ..

It is straightforward to check that ␾ is a bijection. Let ␣ and ␤ be 436 MARK V. LAWSON

X 1 U elements of M . By definition, ␾␣Ž.s Ž␣(␣y,␣ .Ž., ␾␤ s 1 U 1 U Ž␤(␤y,␤.Ž.ŽŽ.Ž.and ␾␣(␤ s ␣(␤(␣(␤y,Ž.␣(␤ .. First, UUU UU Ž.␣(␤s␣(␤, since ␣ ( ␤ is beneath ␣ ( ␤ , and the uniqueness guaranteed by Lemma 7. It remains to show that

111UU1y1 ␣(␤(␤y(␣ys␣(␣y(␣(␤(␤y(Ž.␣,

U 1 U but this follows from the fact that ␣ : ␣ and so ␣ s ␣ ( ␣y ( ␣ s U 1 ␣ ( ␣y ( ␣. X Put F s M _ H. Then F is a semidirect product of H and the semilattice of all co-finite idempotents of IŽރU . except for the idempotent defined on the set ރU itself. Clearly M is an inverse subsemigroup of F. We shall now investigate the nature of this embedding in more detail.

DEFINITION. Let Ž.P, F be a poset. A subset Q of P is said to be an order ideal if x g Q and y F x implies that y g Q. The following is a standard result which we include for the sake of completeness.

LEMMA 10. LetMbeanin¨erse subsemigroup of an in¨erse semigroup F. If the set of idempotents of M is an order ideal of the set of idempotents of F, then M is an order ideal of F.

1 1 1 Proof. Let x g M and y F x. Then yy y F xy x. But xy x is an idempotent in M so by assumption, yy1 y is also an idempotent in M. But 1 y s xyy y. Thus y is an element of M. The next definition is taken fromwx 4 . It will enable us to describe precisely the relationship between M and F.

DEFINITION. Let M be an inverse subsemigroup of an inverse semigroup F. Then F is said to be an enlargement of M if the following three conditions hold: Ž.E1 M is an order ideal of F. 1 1 Ž.E2 If a g F and aay ,aayg M then a g M. Ž.E3 Every idempotent of F is D-related to an idempotent of M.

LEMMA 11. Let M be an order ideal and in¨erse subsemigroup of an in¨erse semigroup F. Suppose that e is an idempotent of F and a is an element 1 1 of F such that e s aay and ay a g M. If f F e, then there is an element 1 1 b g F such that f s bby and by b g M. 1 1 1 Proof. Put b s fa. Then bby s f and by b F ay a. But M is an 1 order ideal of F so that bby gM. THE MOBIUS¨ INVERSE MONOID 437

PROPOSITION 12. F is an enlargement of M. Proof. ConditionŽ. E1 holds. Clearly any idempotent of F beneath an idempotent of M is also in M. The result now follows from Lemma 10. U ConditionŽ. E2 holds. Let ␣ satisfy the assumptions. Now ␣ : ␣ and 1 U 1 1 1 so ␣ s Ž␣ ( ␣y .( ␣ (Ž␣y ( ␣ .. By assumption ␣y ( ␣, ␣ ( ␣y g IŽ.ރ . But 1ރ is the identity of IŽ.ރ . Thus

y1 U y1 ␣ s Ž.Ž.Ž.␣ ( ␣ ( 1ރރ( ␣ (1 ( ␣ ( ␣ .

U But then by Lemma 4Ž. i , 1ރރ( ␣ (1 is a Mobius¨ transformation. Thus ␣gIŽ.ރ. ConditionŽ. E3 holds. The maximal idempotents of F are defined on U sets of the form ރ _ Ä4w where w g ރ,orރ. By Lemma 8, every U idempotent defined on a domain of the form ރ _ Ä4w is D-related to 1ރ . By conditionŽ. E1 and Lemma 11, it is easy to see that every idempotent of F is also D-related to an idempotent of M.

DEFINITION.AMcAlister tripleŽ. G, X, Y consists of a group G, a poset X, and an order ideal Y of X, which is also a meet semilattice, satisfying the axioms: Ž.MT1 G acts on X by order automorphisms. Ž.MT2 GY s X. Ž.MT3 gY l Y is non-empty for all g g G. For the proof of the following seewx 6 .

THEOREM 13. LetŽ. G, X, Y be a McAlister triple. Let

1 P s PGŽ.Ž.,X,Y sÄ4e,g gY=G:gyegY with the multiplication

Ž.Ž.Že, gf,hsengf , gh ..

Then P is an E-unitary in¨erse semigroup. The following result will provide us with all we need to describe the structure of the Mobius¨ inverse monoid.

THEOREM 14. Let F be a semidirect product of a meet semilattice X by a group H. Let M be an in¨erse subsemigroup of F such that F is an enlargement of F. Put Y s Ä y g X: Ž.y,1 gEM Ž.4.ThenŽ. H, X, Yisa McAlister triple and M s PHŽ.,X,Y. 438 MARK V. LAWSON

1 Proof. Recall that in a semidirect product, we have that Ž.e, g ys 1 1 1 1 1 Ž gxy ,gy .Ž,x,gx .yŽ.Ž,gsgxy, 1. , and Žx, gx .Ž,g .y sŽ.x,1 . We show first that Y is an order ideal of X. Let y g Y and suppose that xFy. Then Ž.Ž.x,1 F y,1 in F. But Ž.y,1 gM. Thus Ž.x,1 gM by Ž. E1 . Hence x g Y. To show thatŽ. MT2 holds, let x g X. Then Ž.Ž.Ž.x,1 gE F .ByE3,we have that Ž.Ž.x,1 D y, 1 for some Ž.y,1 gE Ž.M . Thus there exists Žz, g .g 1 1 Fsuch that Ž.Ž.z, gz,gysŽ.x, 1 and Žz, gz .yŽ.Ž.,gsy, 1 . Thus x s z 1 and y s gy z. Hence x s gy. To show thatŽ. MT3 holds, let g g H and let x g X and y g Y be any 1 elements. Then Ž.Ž.Ž.y,1 x, gy,1 gM by Ž. E2 . Thus gyyŽ.nxngy and X X 1 X y n x n gy both belong to Y. Put y s y n x n gy. Then y , gy y g Y. Finally, it is immediate fromŽ. E2 that Žx, g .g M m Žx, g .g PHŽ.,X,Y. The following is now immediate from Proposition 12 and Theorem 14.

THEOREM 15. Let H be the group of extended Mobius¨ transformations defined on ރU, and let X be the partially ordered set of all co-finite idempotents of IŽރU . excluding the idempotent defined on ރU itself. Let H act on X 1 by g.e s gegy . Let Y be the subset of X consisting of the co-finite idempotents of IŽ.ރ . Then Ž H, X, Y . is a McAlister triple and M is isomorphic to the in¨erse semigroup PŽ. H, X, Y .

REFERENCES

1. J. W. Archbold, ‘‘Algebra,’’ Pitman, London, 1970. 2. J. M. Howie, ‘‘An Introduction to Semigroup Theory,’’ Academic Press, San Diego, 1976. 3. M. V. Lawson, Almost factorisable inverse semigroups, Glasgow Math. J. 36 Ž.1994 , 97᎐111. 4. M. V. Lawson, Enlargements of regular semigroups, Proc. Edinburgh Math. Soc. 39 Ž.1996 , 425᎐460. 5. D. Pedoe, ‘‘Geometry,’’ Dover, New York, 1988. 6. M. Petrich, ‘‘Inverse Semigroups,’’ Wiley, New York, 1984. 7. J. Roe, ‘‘Elementary Geometry,’’ Oxford Univ. Press, London, 1993.