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,…,xnx1,…,xn defined by homogeneous relations of the form xi1xi2⋯xil=xσ(i1)xσ(i2)⋯xσ(il)xi1xi2⋯xil=xσ(i1)xσ(i2)⋯xσ(il), where l≥2l≥2 is a given integer and σ runs through a subgroup H   of SymnSymn, 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)Sn,l(H) is a submonoid of its universal group G. If, furthermore, H   is transitive then the periodic elements T(G)T(G) of G form a finite abelian subgroup, G   is periodic-by-cyclic and it is a central localization of Sn,l(H)Sn,l(H), and the Jacobson radical of the algebra A   is determined by the Jacobson radical of the group algebra K[T(G)]K[T(G)]. Finally, it is shown that if H   is an arbitrary group that is transitive then K[Sn,l(H)]K[Sn,l(H)] is a Noetherian PI-algebra of Gelfand–Kirillov dimension one; if furthermore H   is abelian then often K[G]K[G] is a principal ideal ring. In case H   is not transitive then K[Sn,l(H)]K[Sn,l(H)] is of exponential growth.