Ž . JOURNAL OF ALGEBRA 200, 428438 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 showthat 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.  1998 Academic Press 1. INTRODUCTION Mobius transformations can be handled in two ways. The naive way ¨ views them as partial functions defined on the complexplane, 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 Mobius transforma- ¨ tions from the naive standpoint. Formally, a Mobius transformation is a partial function  of the complex ¨ Ž. Ž . Ž . plane, , having the following form:  z  az  b  cz  d where a, b, c, d are complexnumbers and ad  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 func- tion. 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  d  0. Thus the transformations are of two types: the functions in which c  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 dc and the image omits the point ac. The condition ad  bc  0 ensures that in both cases the transformations are injective. Thus Mobius transformations are either ¨ bijections or partial bijections of . The inverse transformation is also a 1 Ž. Ž . Ž . Mobius transformation, namely  z  dz  b cz  a . ¨ 428 0021-869398 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.