Let R be a commutative noetherian ring. Denote by $${\textsf{mod }}\,R$$ the category of finitely generated R-modules. In this paper, we study n-torsionfree modules in sense Auslander and Bridger, comparing them with n-syzygy modules, satisfying Serre’s condition $$(\mathrm {S}_n)$$ . We mainly investigate closedness properties full subcategories consisting those modules.

A permutation group G?Sym(?) is said to be 2-closed if no H such that G<H?Sym(?) has the same orbits on ?×? as G. simple and efficient inductive criterion for 2-closedness established abelian groups with cyclic transitive constituents.

We introduce stochastic integer programs with dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-structured, mixed-integer linear programming equivalents to the dominance constrained stochastic programs are id...