Finitely presented monoids and algebras defined by permutation relations of abelian type

The class of finitely presented algebras over a field K with a set of generators a1,…,an and defined by homogeneous relations of the form a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), where σ runs through an abelian subgroup H of , the symmetric group, is considered. It is proved that the Jacobson radical of such algebras is zero. Also, it is characterized when the monoid Sn(H), with the “same” presentation as the algebra, can be embedded in a group in terms of the stabilizer of 1 and the stabilizer of n in H, where H is an arbitrary subgroup of . This work is a continuation of earlier work of Cedó, Jespers and Okniński.