Finitely presented algebras and groups defined by permutation relations

The class of finitely presented algebras over a field K with a set of generators a₁,…,a_n and defined by homogeneous relations of the form a₁a₂?a_n=a_σ(1)a_σ(2)?a_σ(n), where σ runs through a subset H of the symmetric group Sym_n of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Sym_n of order n. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Sym_n are proposed.