Finitely Presented Monoids and Algebras defined by Permutation Relations of Abelian Type, II

The class of finitely presented algebras A over a field K with a set of generators x1, . . . , xn and defined by homogeneous relations of the form xi1xi2 · · · xil = xσ(i1)xσ(i2) · · · xσ(il), where l ≥ 2 is a given integer and σ runs through a subgroup H of Symn, is considered. It is shown that the underlying monoid Sn,l(H) = 〈x1, x2, . . . , xn | xi1xi2 · · · xil = xσ(i1)xσ(i2) · · · xσ(il), σ ∈ H, i1, . . . , il ∈ {1, . . . , n}〉 is cancellative if and only if H is semiregular and abelian. In this case Sn,l(H) is a submonoid of its universal groupG. 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 Sn,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[Sn,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[Sn,l(H)] is of exponential growth.