Algebraic structures among virtual singular braids

TL;DR

This paper introduces the virtual singular braid group, embedding the virtual singular braid monoid into a group structure, and provides presentations and decompositions that facilitate understanding its algebraic properties.

Contribution

It establishes the embedding of the virtual singular braid monoid into a new group, describes its structure as a semi-direct product, and provides a presentation for the pure subgroup.

Findings

VSG_n is a group containing the monoid as a subset.

VSG_n decomposes into a semi-direct product of pure braids and symmetric group.

A normal form for words in VSG_n is derived.

Abstract

We show that the virtual singular braid monoid on $n$ strands embeds in a group $VSG_n$, which we call the virtual singular braid group on $n$ strands. The group $VSG_n$ contains a normal subgroup $VSPG_n$ of virtual singular pure braids. We show that $VSG_n$ is a semi-direct product of $VSPG_n$ and the symmetric group $S_n$. We provide a presentation for $VSPG_n$ via generators and relations. We also represent $VSPG_n$ as a semi-direct product of $n-1$ subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.