International Journal of Algebra and Statistics Volume 1: 2(2012), 1–7 Published by Modern Science Publishers Available at: http://www.m-sciences.com Pure Fuzzy Ideals in Ternary Semigroups Shahida Bashir a , Muhammad Shabir b , Noor Rehman b a Department of Mthematics University of Gujrat, Gujrat, Pakistan. b Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. (Received: 30 April 2012; Accepted: 19 June 2012) Abstract. In this paper we initiate the study of pure fuzzy ideals in ternary semigroups. We introduce the notions of purely fuzzy maximal and purely fuzzy prime ideals of a ternary semigroup T. We also characterize those ternary semigroups for which each fuzzy ideal is weakly pure. 1. Introduction Ternary semigroups introduced by Lehmer in 1932 in [4]. Sioson developed ideal theory of ternary semigroups in his paper [10]. Many researcher worked on ternary semigroups (see [7–9] ). The fuzzy set was initiated by Zadeh [11]. This theory has generated great enthusiasm among intellectuals specially researchers, working in multilateral aspects of mathematics. Many concepts of Mathematics are applied to such sets and various qualities of these concepts in the contexts of fuzzy sets are established. Many examples like Rosenfeld [6] utilized this idea to delineate some essential primary concepts of groupoids. Further example of such a generalization is Kuroki’s classical semigroup [3]. Some papers by Zadeh which are published afresh in [12] along with his deeply established contributions to the domain of fuzzy sets and their modelings near to reasoning. To solve complicated social and biological problems new tools ought to be devised. Such a mathematics will be competent to handle indeterminacies taking decisions and designing large systems and networks which are complicated, irregular and distribu- tive. Let us refer to [13] for an important article by Zadeh depicting a general theory of uncertainty. One of the profusely applied areas where fuzzy sets have been used abundantly is in executing decision making pertaining to management [14]. The monograph by Mordeson et. al. which is a systematic presentation of fuzzy semigroups came to the forefront in [5], where there are theoretical results on fuzzy semigroups and their employment in fuzzy coding, fuzzy finite state machines and fuzzy languages. Fuzziness enjoys a homely place in the domain of formal languages. The notions of pure fuzzy, purely fuzzy maximal and purely fuzzy prime ideals of a semigroup were introduced by Ahsan, et. al. in [1]. The notions of pure ideals of a ternary semigroup T and verification of a ternary semigroup T is right weakly regular if and only if every ideal of T is right pure is introduced by Bashir and Shabir in [2]. 2010 Mathematics Subject Classification. 20N10. Keywords. Ternary semigroup; Pure Fuzzy ideals; weakly regular ternary semigroups. Email addresses: shahida.bashir@uog.edu.pk (Shahida Bashir), mshabirbhatti@yahoo.co.uk (Muhammad Shabir), noorrehman82@yahoo.com (Noor Rehman)