On minimal generators for semi-closed polyhedra

A semi-closed polyhedron is defined as the intersection of finitely many closed and/or open half-spaces, which can be regarded as an extension of a polyhedron. The same as a polyhedron, a semi-closed polyhedron admits both (Formula presented.) -representation and (Formula presented.) -representation. In this paper, we introduce the concept of a minimal (Formula presented.) -representation for semi-closed polyhedra which can be regarded as a natural extension of a minimal (Formula presented.) -representation for polyhedra. We derive some criteria for refining generators of a semi-closed polyhedron. These refining criteria can be applied to obtain a minimal generator for a semi-closed polyhedron.