Order Cancellation Law in a Semigroup of Closed Convex Sets

TL;DR

This paper extends order cancellation laws in semigroups of convex sets by introducing weakly narrow sets and embedding the semigroup into a topological vector space, broadening previous theorems.

Contribution

It generalizes order cancellation laws for convex sets using weakly narrow sets and topological embeddings, extending Radstrom's theorem.

Findings

Order cancellation law holds for weakly narrow sets in normed spaces.

Order cancellation law applies to convex sets with bounded Hausdorff-like distance from their recession cone.

Semigroup of convex sets can be embedded into a topological vector space.

Abstract

In this paper generalize Robinson's version of an order cancellation law for subsets of vector spaces in which we cancel by unbounded sets. We introduce the notion of weakly narrow sets in normed spaces, study their properties and prove the order cancellation law where the canceled set is weakly narrow. Also we prove the order cancellation law for closed convex subsets of topological vector space where the canceled set has bounded Hausdorff-like distance from its recession cone. We topologically embed the semigroup of closed convex sets sharing a recession cone having bounded Hausdorff-like distance from it into a topological vector space. This result extends Bielawski and Tabor's generalization of Radstrom theorem.