  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_{sigma(1)}a_{sigma(2)}...a_{sigma(n)}, where sigma 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, is cancellative in terms of the stabilizer of 1 and the stabilizer of
n in H. This work is a continuation of earlier work of Cedo, Jespers and
Okninski.
