On the structure of the singular triplet monoid and its virtual extension

In this article, we introduce two new algebraic structures associated with the triplet group on $n$ strands, $L_n$: the singular triplet monoid $SLM_n$ and its virtual extension $VSLM_n$, defined in analogy with the singular braid monoid and the virtual singular braid monoid.

We begin by presenting these monoids in terms of generators and relations, and then derive several alternative presentations of $VSLM_n$.

Second, we investigate the problem of extending representations of $L_n$ to these monoids.

Two extension methods are developed: the $k$-local type extension, which applies to $k$-local representations, and the $\Phi$-type extension, which applies to representations satisfying suitable commutativity conditions.

We show that every $2$-local representation of $L_n$ admits extensions to both $SLM_n$ and $VSLM_n$ via the two methods.

As an application, we consider a specific representation $\mu : L_n \longrightarrow \mathrm{GL}_n(\mathbb{Z}[t^{\pm1}])$ introduced recently by Nasser et al.

We explicitly determine all homogeneous $2$-local extensions of $\mu$ to $SLM_n$ and $VSLM_n$, and compute the corresponding $\Phi$-type extensions.

Furthermore, we compare these two extension methods, showing that they coincide for $SLM_n$ under suitable parameter conditions, while they do not coincide for $VSLM_n$.

These results provide a systematic framework for extending representations of $L_n$ to $SLM_n$ and $VSLM_n$.