The class of finitely presented algebras A over a field K with a set of generators x_{ 1},...,x_{ n} and defined by homogeneous relations of the form x_{ i_1}x_{ i_2}...x_{ i_l}=x_{ sigma(i_1)}x_{ sigma(i_2)}...x_{ sigma(i_l)}, where l geq 2 is a given integer and sigma runs through a subgroup H of Sym_n, is considered. It is shown that the underlying monoid S_{ n,l}(H)= <x_1,x_2,...,x_n|x_{ i_1}x_{ i_2}...x_{ i_l}=x_{ sigma(i_1)}x_{ sigma(i_2)}...x_{ \sigma (i_l)}, sigma in H, i_1,...,i_l in { 1,...,n}> is cancellative if and only if H is semiregular and abelian. In this case S_{ n,l}(H) is a submonoid of its universal group G. If, furthermore, H is transitive then the periodic elements T(G) of G form a finite abelian subgroup, G is periodic-by-cyclic and it is a central localization of S_{ n,l}(H), and the Jacobson radical of the algebra A is determined by the Jacobson radical of the group algebra K[T(G)]. Finally, it is shown that if H is an arbitrary group that is transitive then K[S_{ n,l}(H)] is a Noetherian PI-algebra of Gelfand-Kirillov dimension one; if furthermore H is abelian then often K[G] is a principal ideal ring. In case H is not transitive then K[S_{ n,l}(H)] is of exponential growth.
Categories: math.RA