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A note on minimal and maximal ideals of ordered semigroups
Lobachevskii journal of mathematics, Tome 11 (2002), pp. 3-6
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Ideals of ordered groupoids were defined by second author in [2]. Considering the question under what conditions an ordered semigroup (or semigroup) contains at most one maximal ideal we prove that in an ordered groupoid $S$ without zero there is at most one minimal ideal which is the intersection of all ideals of $S$. In an ordered semigroup, for which there exists an element a $\in S$ such that the ideal of $S$ generated by $a$ is $S$, there is at most one maximal ideal which is the union of all proper ideals of $S$. In ordered semigroups containing unit, there is at most one maximal ideal which is the union of all proper ideals of $S$. 
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M. M. Arslanov; N. Kehayopulu. A note on minimal and maximal ideals of ordered semigroups. Lobachevskii journal of mathematics, Tome 11 (2002), pp. 3-6. http://geodesic.mathdoc.fr/item/LJM_2002_11_a0/

[1] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll, Publ., XXV, Providence, 1967 | MR

[2] N. Kehayopulu, “On weakly prime ideals of ordered semigroups”, Mathematica Japonica, 35:6 (1990), 1051–1056 | MR | Zbl

[3] N. Kehayopulu, “Note on Green's relations in ordered semigroups”, Mathematica Japonica, 36:2 (1991), 211–214 | MR | Zbl