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 a 1,...,a n and defined by homogeneous relations of the form a 1a 2⋯a n=a σ(1)a σ(2)⋯a σ(n), where σ runs through an abelian subgroup H of Sym n, the symmetric group, is considered. It is proved that the Jacobson radical of such algebras is zero. Also, it is characterized when the monoid S n(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 Sym n. This work is a continuation of earlier work of Cedó, Jespers and Okniński. © 2011 Elsevier B.V.