(untitled) (European Journal of Pure and Applied Mathematics, 2018)
For a proper subset A of an ordered semigroup S, we denote by H_A(S) the subset of S defined by H_A(S):={hS such that if sSA, then s otin (shS]}. We prove, among others, that if A is a right ideal of S and the set H_A(S) is nonempty, then H_A(S) is an ideal of S; in particular it is a semiprime ideal of S. Moreover, if A is an ideal of S, then AH_A(S). Finally, we prove that if A and I are right ideals of S, then IH_A(S) if and only if s otin (sI] for every sSA. We give some examples that illustrate our results. Our results generalize the Theorem 2.4 in Semigroup Forum 96 (2018), 523–535.