The relation <_ is defined on the set of right ideals of an ordered semigroup. The main result of this paper is as follows: an ordered semigroup S is a chain of right simple ordered semigroups if and only if < is an order relation. Bibliography: 3 titles.In [3] B. Pond61i6ek has shown that an associative ring A is a division ring if and only if the right ideals of A form a chain with respect to the relation P < Q -.' ~-P = PQ = QP. He also proved that a semigroup S is a chain of right simple semigroups if and only if the right ideals of S form a chain with respect to the relation P < Q r P = PQ = QP (see Theorem 2 in [3]). In this paper we give an analog of Theorem 2 from [3] for ordered semigroups.A po-semigroup (an ordered semigroup) is an ordered set S that is a semigroup such that a < b ==c, xa <_ xb and ax < bx Vx e S (see [I]). ForA, BCS, letAB:= {abIaEA, bEB}.
Definition 1. Let S be a po-semigroup and ~ ~ A C S. The set A is called aright (resp., left) ideal orS if (1) AS C A (resp., SAC A);(2) aEA, SBb
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.