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Finitely presented algebras defined by permutation relations of dihedral type

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},\ldots , a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{σ(1)} a_{σ(2)} \cdots a_{σ(n)}$, where $σ$ runs through a subset $H$ of the symmetric group $\text{Sym}_{n}$ of degree $n$, is investigated. Groups $H$ in which the cyclic group $\langle (1,2, \ldots ,n) \rangle$ is a normal subgroup of index $2$ are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid $S_n(H)$ with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovski\uı. The universal group $G_n$ of $S_n(H)$ is a unique product group, and it is the central localization of a cancellative subsemigroup of $S_n(H)$. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra $K[S_n(H)]$ is semiprimitive.