On PM-monoids and braid PM-monoids

TL;DR

This paper introduces PM-monoids and braid PM-monoids, exploring their structures, geometric interpretations, and solving the word problem for braid PM-monoids, expanding the algebraic understanding of these monoids.

Contribution

It defines new monoids containing symmetric and braid groups, develops their theory, and provides solutions to their word problems.

Findings

Braid PM-monoids are described by geometric braids.

The structure of PM-monoids is characterized via matched pairs.

A solution to the word problem for braid PM-monoids is established.

Abstract

In this paper, we shall introduce two monoids. One is called a PM-monoid which contains the symmetric group, the other is called a braid PM-monoid which contains the braid group. We shall develop the theory of PM-monoids and that of braid PM-monoids. The PM-monoids is obtained in the context of the compactification of projective linear group defined by Mutsumi Saito. The structure of PM-monoids is described in terms of matched pairs. We can define braid PM-monoid using a presentation for the PM-monoid. As main results, we show that braid PM-monoids are described by geometric braids and we find a solution to the word problem for the braid PM-monoids.