The monoid structure of singular twisted virtual braids

In this paper, we examine specific submonoids within the singular twisted virtual braid monoid $STV{B}_n$. Notably, we establish that the singular twisted virtual pure braid monoid $STV{P}_n$ serves as the kernel of an epimorphism from $STV{B}_n$ onto the symmetric group $S_n$. We identify the generators and defining relations for $STV{P}_n$. Additionally, we construct other epimorphisms from $STV{B}_n$ onto $S_n$, whose kernels are analogous to $STV{P}_n$, and determine their respective generators and defining relations. Furthermore, we demonstrate the embedding of the monoid $STV{B}_n$ into a group. Also, we provide the extension of the representation of the twisted virtual braid group to the representation of the singular twisted virtual braid monoid.