Factorisations of finitely presented special inverse monoids

Abstract: Inverse monoids are a kind of algebraic object which model partial bijections in the same way that groups model bijections and semigroups model transformations generally. A special inverse monoid is one defined by a set of relations of the form $r_i = 1$. We can factorise these $r_i$ into component words, for instance $xyxzxy =1$ may be written $(xy)(xz)(xy)=1$. In particular, every special inverse monoid presentation has a unique factorisation of its relators into minimal invertible pieces.

In 2001 Ivanov, Margolis and Meakin produced a paper which included amongst others two important results. That the special inverse monoid's group of units is generated by the minimal invertible pieces and that we can decide the word problem of an E-unitary special inverse monoid by deciding the word and prefix membership problems of its maximal group image. Both allow us to better understand an inverse monoid by studying corresponding groups.

In this talk we will use these results to discuss the effects of the minimal invertible pieces of a special inverse monoid's relators having certain combinatorial properties on a number of decision problems, extending results by Dolinka and Gray.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.