<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper we generalize a result of R. Urbański from paper [7] which states that for subsets <italic>A</italic>, <italic>B</italic>, <italic>C</italic> of topological vector space <italic>X</italic> the following implication holds 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0026_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>⊂</mo><mi>B</mi><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>A</mi><mo>⊂</mo><mi>C</mi></mrow></math>
<tex-math>A + B \subset B + C \Rightarrow A \subset C</tex-math>
</alternatives>
</disp-formula>
 provided that <italic>B</italic> is bounded and <italic>C</italic> is closed and convex. The generalization is given in Theorem 5 where we prove this result for <italic>k</italic>- convex subsets of a topological vector space. Also we introduce a notion of some abstract like-closure operation for subsets of linear space and we study its connections to order cancellation law.</p>
</abstract>

In this paper we generalize a result of R. Urbański from paper [7] which states that for subsets A, B, C of topological vector space X the following implication holds 



A+B⊂B+C⇒A⊂C
A + B \subset B + C \Rightarrow A \subset C


 provided that B is bounded and C is closed and convex. The generalization is given in Theorem 5 where we prove this result for k- convex subsets of a topological vector space. Also we introduce a notion of some abstract like-closure operation for subsets of linear space and we study its connections to order cancellation law.

On some generalization of order cancellation law for subsets of topological vector space

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

In this paper we generalize a result of R. Urbański from paper [7] which states that for subsets A, B, C of topological vector space X the following...

On some generalization of order cancellation law for subsets of topological vector space

Published Online: Sep 29, 2023

Page range: 378 - 382

Received: Jun 27, 2023

Accepted: Sep 27, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0026

Keywords
convex sets, Minkowski addition of sets, cancellation law

© 2023 Hubert Przybycie, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On some generalization of order cancellation law for subsets of topological vector space

Published Online: Sep 29, 2023

Page range: 378 - 382

Received: Jun 27, 2023

Accepted: Sep 27, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0026

Keywordsconvex sets, Minkowski addition of sets, cancellation law

© 2023 Hubert Przybycie, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
convex sets, Minkowski addition of sets, cancellation law