We introduce the notion of ordered quasi- -ideals of regular ordered -semigroups and study the basic properties of ordered quasi- -ideals of ordered -semigroups. We also characterize regular ordered -semigroups in terms of their ordered quasi- -ideals, ordered right -ideals, and left -ideals. Finally, we have shown that (i) a partially ordered -semigroup is regular if and only if for every ordered bi- -ideal , every ordered -ideal , and every ordered quasi- -ideal , we have and (ii) a partially ordered -semigroup is regular if and only if for every ordered quasi- -ideal , every ordered left -ideal , and every ordered right- -ideal , we have that . 1. Introduction Steinfeld [1–3] introduced the notion of a quasi-ideal for semigroups and rings. Since then, this notion has been the subject of great attention of many researchers and consequently a series of interesting results have been published by extending the notion of quasi-ideals to -semigroups, ordered semigroups, ternary semigroups, semirings, -semirings, regular rings, near-rings, and many other different algebraic structures [4–15]. It is a widely known fact that the notion of a one-sided ideal of rings and semigroups is a generalization of the notion of an ideal of rings and semigroups and the notion of a quasi-ideal of semigroups and rings is a generalization of a one-sided ideal of semigroups and rings. In fact the concept of ordered semigroups and -semigroups is a generalization of semigroups. Also the ordered -semigroup is a generalization of -semigroups. So the concept of ordered quasi-ideals of ordered semigroups is a generalization of the concept of quasi-ideals of semigroups. In the same way, the notion of an ordered quasi-ideal of ordered semigroups is a generalization of a one-sided ordered ideal of ordered semigroups. Due to these motivating facts, it is naturally significant to generalize the results of semigroups to -semigroups and of -semigroups to ordered -semigroups. In 1998, the concept of an ordered quasi-ideal in ordered semigroups was introduced by Kehayopulu [16]. He studied theory of ordered semigroups based on ordered ideals analogous to the theory of semigroups based on ideals. The concept of po- -semigroup was introduced by Kwon and Lee in 1996 [17] and since then it has been studied by several authors [18–22]. Our purpose in this paper is to examine many important classical results of ordered quasi- -ideals in ordered -semigroups and then to characterize the regular ordered -semigroups through ordered quasi- -ideals, ordered bi- -ideals and ordered one-sided -ideals. 2.
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