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

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.