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The Mobius Inverse Monoid JOURNAL OF ALGEBRA 200, 428]438Ž. 1998 ARTICLE NO. JA977242 The MobiusÈ 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 MobiusÈ transformations generate an F-inverse monoid whose maximum group image is the MobiusÈ group. We describe the monoid in terms of McAlister triples. Q 1998 Academic Press 1. INTRODUCTION MobiusÈ 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 com- plex plane. In the first part of this paper, I shall view MobiusÈ transforma- tions from the naive standpoint. Formally, a MobiusÈ transformation is a partial function a of the complex plane, C, having the following form: aŽ.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 a because multiplying a, b, c, and d by a non-zero complex number results in a different form but the same func- tion. This must always be borne in mind in the sequel. MobiusÈ 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 MobiusÈ transformations are either bijections or partial bijections of C. The inverse transformation is also a 1 MobiusÈ transformation, namely ay Ž.z s Ždz y b .Žrycz q a .. 428 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. THE MOBIUSÈ INVERSE MONOID 429 MobiusÈ transformations can be ``multiplied.'' If aŽ.z s Žaz q b .Žr cz q d.Ž.Ž.Ž.and b z s Az q B r Cz q D are MobiusÈ transformations then their product, which we shall denote by a(b, 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 MobiusÈ transformation, as can easily be checked, consequently ( is a well-defined binary operation on the set of MobiusÈ transformations. With respect to this operation the MobiusÈÈ 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 MobiusÈ transformation as associated withwx a matrix, then it is easily checked that the effect of composing two MobiusÈ transformations is to multiply the corresponding matrices. The composite of two MobiusÈ transforma- tions is therefore another MobiusÈ transformation. To see why these comments are misleading, consider the following two specific transformations: aŽ.z s Ž2 z q 1 .Žr z y 1 . and b Ž.z s Ž2 z q 1.Žrzq1 . As partial functions a: C _ Ä41 ª C _ Ä42 and b: C _yÄ1 4ª C_Ä42 . We compute their composition, a ( b, as partial functions. Ob- 1 serve that domŽ.a l im Žb .s C _ Ä41, 2 and so domŽ.a ( b s byŽC _ Ä41, 2 .s C _ Ä40, y1 and imŽ.Ža ( b s a C _ Ä41, 2 .s C _ Ä42, 5 . We see that the partial function composition of a and b is not a MobiusÈ transformation, because two points are omitted from the domain of the resulting function. Thus the composition of two MobiusÈ 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 MobiusÈ transformations can be derived in a natural way from their composition. To do this, we consider the inverse monoid generated by the MobiusÈ transformations regarded as partial functions of C. In the second part of the paper, we analyse the structure of this inverse monoid. 2. THE MOBIUSÈ GROUP VIA INVERSE MONOIDS We shall assume the reader is familiar with elementary inverse semi- group 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 compos- ite, 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 subsemi- group. Remark. We shall refer to the elements of the above inverse semigroup as ``co-finiteŽ. partial bijections .'' Since MobiusÈ transformations are partial bijections of C we can regard them as elements of the symmetric inverse monoid IŽ.C . The finite compositions of MobiusÈ transformations and their inverses generate an inverse submonoid of IŽ.C , which we call the MobiusÈ in¨erse monoid, denoted M. LEMMA 2. The idempotents of M consist of the identity functions defined on the co-finite subsets of C. Proof. Each element of M is a finite composition of MobiusÈ transfor- mations 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.C is an element of M. Let w be any complex number. Every subset of the form C _ Ä4w can be obtained as the domain of some MobiusÈ transformation in which w sydrc. Finally, the identity defined on the subset C _ Ä4w1,...wn can be obtained by composing the identities defined on the subsets C _ Ä4w1 ,...,C_ Ä4wn . The proof of the following is immediate from the definitions. LEMMA 3. a ( b : a(b. The following property of MobiusÈ transformations is fundamental. See wx1 for a proof. THEOREM 4. A MobiusÈ transformation which fixes three or more points is the identity. Hence two MobiusÈ transformations which agree on three or more points must be equal. THE MOBIUSÈ INVERSE MONOID 431 The above result has the following important consequence for the structure of M. LEMMA 5. If a g M and a : b, g where b and g are MobiusÈ transfor- mations, then b s g. Proof. The functions b and g agree on the set domŽ.a . 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 MobiusÈ transformations generate an E-unitary in- ¨erse submonoid M of IŽ.C . The maximal elements of M are the MobiusÈ transformations, and e¨ery element of M lies beneath a unique maximal element. Proof. By Lemma 3, the composition of two MobiusÈ transformations is beneath a MobiusÈ transformation. Thus the composition of any finite number or MobiusÈÈ transformations is beneath a Mobius transformation, and so every element of M is beneath a MobiusÈ transformation. The MobiusÈ transformations form the maximal elements of M with respect to the natural partial order, for if a : b where a is a MobiusÈ transformation and b an arbitrary element of M, then b : g for some MobiusÈ transfor- mation g. Thus a : b. But then by Lemma 5, a s b. We have thus proved that the maximum elements in the poset Ž.M, : are precisely the MobiusÈ 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 s is defined on an in- verse semigroup S by Ž.s, t g s 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 s . 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.
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