Construction of a two unique product semigroup defined by permutation relations of quaternion type

TL;DR

This paper constructs a specific monoid based on permutation relations of quaternion type and proves it has the two unique product property, leading to algebraic properties like being a domain and semiprimitive.

Contribution

It introduces a new class of monoids defined by quaternion-type permutation relations and proves their two unique product property.

Findings

The monoid $S_n(H)$ has the two unique product property.

The monoid algebra $K[S_n(H)]$ is a domain with trivial units.

The algebra is semiprimitive.

Abstract

For a regular representation $H \subseteq \text{Sym}_n$ of the generalized quaternion group of order $n=4k$, with $k\geq 2$, the monoid $S_n(H)$ presented with generators $a_1,a_2,\dots ,a_n$ and with relations $a_1a_2\cdots a_n=a_{\sigma(1)}a_{\sigma(2)}\cdots a_{\sigma(n)}$, for all $\sigma\in H$, is investigated. It is shown that $S_n(H)$ has the two unique product property. As a consequence, for any field $K$, the monoid algebra $K[S_n(H)]$ is a domain with trivial units which is semiprimitive.