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Order Cancellation Law in a Semigroup of Closed Convex Sets

Jerzy Grzybowski, Hubert Przybycien

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.

Order Cancellation Law in a Semigroup of Closed Convex Sets

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.

Paper Structure

This paper contains 6 sections, 36 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Cancellation law in an ordered semigroup with an operator of convergence
  3. Assymptotic and recession cones
  4. Order cancellation law in normed spaces.
  5. Order cancellation law in topological vector spaces
  6. Conclusions

Key Result

Theorem 1.1

(see for example Proposition 2.1 in rU.) Let $X$ be a Hausdorff topological vector space and $A,B,C\subset X$. If $B$ is nonempty and bounded and $C$ is closed and convex then $\textup{cl}(A+B)\subset \textup{cl}(B+C)\Longrightarrow A\subset C$.

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 62 more