# Special Inverse Monoids: Geometry, Structure & Algorithms

Lead Research Organisation:
University of Manchester

Department Name: Mathematics

### Abstract

Groups are one of the biggest success stories of modern mathematics, with a rich and beautiful pure mathematical theory which finds application across and beyond mathematics. In particular, they are well-equipped to model problems which display inherent symmetry or reversibility/invertibility, such as often arise in the physical sciences. Weaker algebraic structures such as semigroups and monoids have an even wider potential domain of application, encompassing problems without global symmetry or reversibility, such as those common in biology and computer science. However, they are harder to prove general theorems about, so there is less pure mathematical knowledge to apply.

Inverse monoids are a special type of monoid which form the natural algebraic model of partial symmetry, that is, similarity between different parts of a structure. They have applications in many areas, including notably functional analysis and mathematical physics. Special inverse monoids are a particular class midway between groups and (general) inverse monoids; although they are not yet well understood, the evidence so far suggests they have a rich and beautiful mathematical structure closely related to that of groups, while also displaying some of the wilder behaviour typical of inverse monoids. They also have applications to some problems of wider interest.

Here is an elementary example: consider words (finite sequences of symbols) over a finite alphabet (set of symbols). Suppose we choose two particular words (u and v say) which we declare to be equivalent. We then regard two longer words as equivalent if we can get from one to the other by repeatedly substituting u for v and vice versa within them. For example, if our rule is that AB is equivalent to BBAB, then AABB will be equivalent (by replacing the AB in the middle with BBAB) to ABBABB, and hence also (by replacing to AB at the beginning with BBAB) to BBABBABB and so forth. An natural question is: is there an algorithm which (for a particular rule) will tell us when two words are equivalent? This problem is called the word problem for one-relator monoids (or just the one-relator word problem); it has been a subject of research for over a century, but we still don't know the answer. However, some relatively recent research of Ivanov, Margolis and Meakin shows that it can be rephrased as a problem about special inverse monoids; since the latter have a very much richer algebraic and geometric structure than monoids in general, this is a promising line of attack.

The project seeks to develop our pure mathematical (algebraic, geometric and algorithmic) understanding of special inverse monoids. As well as potentially enabling progress on problems such as that above, it will act as a stepping stone to a better understanding of more general inverse monoids, and hence ultimately contribute to an algebraic understanding of non-reversible and non-symmetric structures wherever they occur.

Inverse monoids are a special type of monoid which form the natural algebraic model of partial symmetry, that is, similarity between different parts of a structure. They have applications in many areas, including notably functional analysis and mathematical physics. Special inverse monoids are a particular class midway between groups and (general) inverse monoids; although they are not yet well understood, the evidence so far suggests they have a rich and beautiful mathematical structure closely related to that of groups, while also displaying some of the wilder behaviour typical of inverse monoids. They also have applications to some problems of wider interest.

Here is an elementary example: consider words (finite sequences of symbols) over a finite alphabet (set of symbols). Suppose we choose two particular words (u and v say) which we declare to be equivalent. We then regard two longer words as equivalent if we can get from one to the other by repeatedly substituting u for v and vice versa within them. For example, if our rule is that AB is equivalent to BBAB, then AABB will be equivalent (by replacing the AB in the middle with BBAB) to ABBABB, and hence also (by replacing to AB at the beginning with BBAB) to BBABBABB and so forth. An natural question is: is there an algorithm which (for a particular rule) will tell us when two words are equivalent? This problem is called the word problem for one-relator monoids (or just the one-relator word problem); it has been a subject of research for over a century, but we still don't know the answer. However, some relatively recent research of Ivanov, Margolis and Meakin shows that it can be rephrased as a problem about special inverse monoids; since the latter have a very much richer algebraic and geometric structure than monoids in general, this is a promising line of attack.

The project seeks to develop our pure mathematical (algebraic, geometric and algorithmic) understanding of special inverse monoids. As well as potentially enabling progress on problems such as that above, it will act as a stepping stone to a better understanding of more general inverse monoids, and hence ultimately contribute to an algebraic understanding of non-reversible and non-symmetric structures wherever they occur.

### Organisations

## People |
## ORCID iD |

Mark Kambites (Principal Investigator) |