Mathematical Sciences: Presentations of Inverse Monoids

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This research is concerned with the development of a theory of presentations of inverse monoids. There are two general areas of problems which will be studied. The first is concerned with basic decision problems which arise when an inverse monoid is presented by generators and relations. In particular, the word problem and the E-unitary problem will be studied, with initial emphasis focussed on the one relator case. Many of the geometric techinques from combinatorial group theory as well as basic results from language theory and automata theory will be used. The second problem is concerned with a study of the closed inverse submonoids of the free inverse monoid and related problems. In particular, the relationship between the closed inverse submonoids of a free inverse monoid, the topology of finite graphs, the rational and recognizable subsets of a free inverse monoid and the dot depth of a finite aperiodic inverse monoid will be studied in detail. In addition, an attempt will be made to begin a computational theory of finite inverse semigroups of partial one-one transformations. This research concerns the theory of monoids, the simplest of the abstract algebraic objects introduced in the last century. The particular ones that the investigators are interested in are precisely those structures arising in automata theory and the theory of formal languages in computer science. Their techniques range from group theory to geometry to computations. Their approaches to the problems are imaginative. This proposal should make important contributions to both mathematics and computer science.