Igor Dolinka: Recent developments in combinatorial inverse semigroup theory

Recent developments in combinatorial inverse semigroup theory

Igor Dolinka, University of Novi Sad, Serbia

Abstract:

In early 1930s, W. Magnus proved his famous and celebrated result that the word problem is decidable for all one-relator groups. This early major result of combinatorial group theory motivated the consideration of the semigroup-theoretical analogue of the Magnus problem: Is the word problem decidable for all one-relator monoids? Despite almost a century of great efforts, this problem still remains open.

In the second half of the XX century, S. I. Adian and his collaborators solved a significant chunk of special cases of the one-relator monoid problem, thus reducing it to two special cases. In 2001, Margolis and Meakin published a seminal paper establishing a connection between the Adian problem and one-relator inverse monoids. This gave a vital impetus to the development of the combinatorial inverse semigroup theory, in particular to the study of finitely presented special inverse monoids.

In my talk I am going to provide a detailed historical account of these results and connections, together with an overview of recent results in the area obtained in the last few years in collaboration with R. D. Gray.

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Primož Moravec