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

For a regular representation H⊆Symn of the generalized quaternion group of order n=4k, with k≥2, the monoid Sn(H) presented with generators a1,a2,…,an and with relations a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), for all σ∈H, is investigated. It is shown that Sn(H) has the two unique product property. As a consequence, for any field K, the monoid algebra K[Sn(H)] is a domain with trivial units which is semiprimitive.