On the virtual singular braid monoid

TL;DR

This paper investigates the algebraic structure of the virtual singular braid monoid and its pure form, providing a representation and exploring their relationships with symmetric groups.

Contribution

It introduces the algebraic structures of $VSB_n$ and $VSP_n$, and constructs a representation of $VSB_n$, advancing understanding of virtual singular braids.

Findings

Defined the virtual singular braid monoid and pure braid monoid

Established $VSB_n$ as an extension of $VSP_n$ by $S_n$

Constructed a representation of $VSB_n

Abstract

We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct a representation of $VSB_n$.