Let m, n be non-negative integers. A subsemigroup A of an ordered semigroup (S, · , ≤ is called an (m, n)-ideal of S if (i) Am SAn ⊆ A, and (ii) if x ∈ A, y ∈ S such that y ≤ x, then y ∈ A. In this paper, necessary and sufficient conditions for every (m, n)-ideal (resp. (m, n)-quasi-ideal) of an (m, n)-ideal (resp. (m, n)-quasi-ideal) A of S is an (m, n)-ideal (resp. (m, n)-quasiideal) of S will be given. Moreover, (m, n)-regularity of S will be discussed. The results obtained extend the results on semigroups (without order) studied by Bogdanovic′ (1979).
