On quasi-prime, weakly quasi-prime left ideals in ordered-I-semigroups

In this paper we obtain and establish some important results in ordered I-semigroups extending and generalizing those for semigroups given in [PETRICH, M.: Introduction to Semigroups, Merill, Columbus, 1973] and for ordered semigroups from [KEHAYOPULU, N.: On weakly prime ideals of ordered semigroups, Math. Japon. 35 (1990), 1051-1056], [KEHAYOPULU, N.: On prime, weakly prime ideals in ordered semigroups, Semigroup Forum 44 (1992), 341-346] and [XIE, X. Y.-WU, M. F.: On quasi-prime, weakly quasi-prime left ideals in ordered semigroups, PU.M.A. 6 (1995), 105-120]. We introduce and give some characterizations about the quasi-prime and weakly quasi-prime left ideals of ordered-I-semigroups. We also introduce the concept of weakly m-systems in ordered I-semigroups and give some characterizations of the quasi-prime and weakly quasi-prime left ideals by weakly m-systems.