{"title":"On Completely Semiprime, Semiprime and Prime Fuzzy Ideals in Ordered Semigroups","authors":"Jian Tang","volume":51,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":360,"pagesEnd":366,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/15343","abstract":"
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.<\/p>\r\n","references":"[1] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(1965), 338-353.\r\n[2] N. Kehayopulu and M. Tsingelis, Fuzzy sets in ordered groupoids,\r\nSemigroup Forum, 65(2002), 128-132.\r\n[3] N. Kehayopulu and M. Tsingelis, The embedding of an ordered groupoid\r\ninto a poe-groupoid in terms of fuzzy sets, Information Sciences,\r\n152(2003), 231-236.\r\n[4] N. Kehayopulu and M. Tsingelis, Fuzzy interior ideals in ordered\r\nsemigroups, Lobachevskii Journal of Mathematics, 21(2006), 65-71.\r\n[5] N. Kehayopulu and M. Tsingelis, Regular ordered semigroups in terms\r\nof fuzzy subsets, Information Sciences, 176(2006), 3675-3693.\r\n[6] N. Kehayopulu and M. Tsingelis, Fuzzy bi-ideals in ordered semigroups,\r\nInformation Sciences, 171(2005), 13-28.\r\n[7] N. Kehayopulu, On prime, weakly prime ideals in ordered semigroups,\r\nSemigroup Forum, 44(1992), 341-346.\r\n[8] N. Kehayopulu, On weakly prime ideals in ordered semigroups, Math.\r\nJaponica, 35(6)(1990), 1051-1056.\r\n[9] N. Kehayopulu, On intra-regular ordered semigroups, Semigroup Forum,\r\n46(1993), 271-278.\r\n[10] N. Kehayopulu, On regular, intra-regular ordered semigroups, Pure\r\nMath. and Appl., 4(4)(1993), 447-461.\r\n[11] N. Kehayopulu, Note on Green-s relations in ordered semigroups, Math.\r\nJaponica, 36(2)(1991), 211-214.\r\n[12] N. Kehayopulu and M. Tsingelis, On left regular ordered semigroups,\r\nSoutheast Asian Bull. Math., 25(2002), 386-394.\r\n[13] N. Kuroki, On fuzzy semigroups, Information Sciences, 53(1991), 203-\r\n236.\r\n[14] Y.L. Cao, On weakly commutativity of po-semigroups and their semilattice\r\ndecompositions, Semigroup Forum, 58(1999), 386-394.\r\n[15] X. Y. Xie and J. Tang, Fuzzy radicals and prime fuzzy ideals of ordered\r\nsemigroups, Information Sciences, 178(2008), 4357-4374.\r\n[16] X. Y. Xie, J. Tang and F. Yan, A characterization of prime fuzzy ideals of\r\nordered semigroups, Fuzzy Systems and Mathematics, 22(2008), 39-44.\r\n[17] X. Y. Xie and J. Tang, Regular ordered semigroups and intra-regular\r\nordered semigroups in terms of fuzzy subsets, Iranian J. Fuzzy Systems,\r\n7(2)(2010), 121-140.\r\n[18] X. Y. Xie and M. F. Wu, On congruences on ordered semigroups, Math.\r\nJaponica, 45(1997), 81-84.\r\n[19] X. Y. Xie, An introduction to ordered semigroup theory. Beijing: Kexue\r\nPress, 2001.\r\n[20] X. Y. Xie and M. F. Wu, The Theory of Fuzzy Semigroups. Beijing:\r\nKexue Press, 2005.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 51, 2011"}