On Completely Semiprime, Semiprime and Prime Fuzzy Ideals in Ordered Semigroups

In this paper, we first introduce the new concept of completely semiprime fuzzy ideals of an ordered semigroup S, which is an extension of completely semiprime ideals of ordered semigroup S, and investigate some its related properties. Especially, we characterize an ordered semigroup that is a semilattice of simple ordered semigroups in terms of completely semiprime fuzzy ideals of ordered semigroups. Furthermore, we introduce the notion of semiprime fuzzy ideals of ordered semigroup S and establish the relations between completely semiprime fuzzy ideals and semiprime fuzzy ideals of S. Finally, we give a characterization of prime fuzzy ideals of an ordered semigroup S and show that a nonconstant fuzzy ideal f of an ordered semigroup S is prime if and only if f is twovalued, and max{f(a), f(b)} = inf f((aSb]), ∀a, b ∈ S.

• Jian Tang

• Ordered fuzzy point
• fuzzy left (right) ideal of anordered semigroup
• completely semiprime fuzzy ideal
• semiprimefuzzy ideal
• prime fuzzy ideal.

[1] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(1965), 338-353.
[2] N. Kehayopulu and M. Tsingelis, Fuzzy sets in ordered groupoids,
Semigroup Forum, 65(2002), 128-132.
[3] N. Kehayopulu and M. Tsingelis, The embedding of an ordered groupoid
into a poe-groupoid in terms of fuzzy sets, Information Sciences,
152(2003), 231-236.
[4] N. Kehayopulu and M. Tsingelis, Fuzzy interior ideals in ordered
semigroups, Lobachevskii Journal of Mathematics, 21(2006), 65-71.
[5] N. Kehayopulu and M. Tsingelis, Regular ordered semigroups in terms
of fuzzy subsets, Information Sciences, 176(2006), 3675-3693.
[6] N. Kehayopulu and M. Tsingelis, Fuzzy bi-ideals in ordered semigroups,
Information Sciences, 171(2005), 13-28.
[7] N. Kehayopulu, On prime, weakly prime ideals in ordered semigroups,
Semigroup Forum, 44(1992), 341-346.
[8] N. Kehayopulu, On weakly prime ideals in ordered semigroups, Math.
Japonica, 35(6)(1990), 1051-1056.
[9] N. Kehayopulu, On intra-regular ordered semigroups, Semigroup Forum,
46(1993), 271-278.
[10] N. Kehayopulu, On regular, intra-regular ordered semigroups, Pure
Math. and Appl., 4(4)(1993), 447-461.
[11] N. Kehayopulu, Note on Green-s relations in ordered semigroups, Math.
Japonica, 36(2)(1991), 211-214.
[12] N. Kehayopulu and M. Tsingelis, On left regular ordered semigroups,
Southeast Asian Bull. Math., 25(2002), 386-394.
[13] N. Kuroki, On fuzzy semigroups, Information Sciences, 53(1991), 203-
236.
[14] Y.L. Cao, On weakly commutativity of po-semigroups and their semilattice
decompositions, Semigroup Forum, 58(1999), 386-394.
[15] X. Y. Xie and J. Tang, Fuzzy radicals and prime fuzzy ideals of ordered
semigroups, Information Sciences, 178(2008), 4357-4374.
[16] X. Y. Xie, J. Tang and F. Yan, A characterization of prime fuzzy ideals of
ordered semigroups, Fuzzy Systems and Mathematics, 22(2008), 39-44.
[17] X. Y. Xie and J. Tang, Regular ordered semigroups and intra-regular
ordered semigroups in terms of fuzzy subsets, Iranian J. Fuzzy Systems,
7(2)(2010), 121-140.
[18] X. Y. Xie and M. F. Wu, On congruences on ordered semigroups, Math.
Japonica, 45(1997), 81-84.
[19] X. Y. Xie, An introduction to ordered semigroup theory. Beijing: Kexue
Press, 2001.
[20] X. Y. Xie and M. F. Wu, The Theory of Fuzzy Semigroups. Beijing:
Kexue Press, 2005.