Annals of Fuzzy Mathematics and Informatics Volume 9, No. 6, (June 2015), pp. 957–974 ISSN: 2093–9310 (print version) ISSN: 2287–6235 (electronic version) http://www.afmi.or.kr @FMI c  Kyung Moon Sa Co. http://www.kyungmoon.com On prime cubic bi-ideals of semigroups Sadaf Umar, Asmat Hadi, Asghar Khan Received 26 June 2014; Revised 20 November 2014; Accepted 12 December 2014 Abstract. As we know that fuzzy sets initiated by Zadeh in 1965, have several important extensions and generalizations, e.g., intuitionis- tic fuzzy sets, L-fuzzy sets, bipolar fuzzy sets, interval-valued fuzzy sets and n-dimensional fuzzy sets etc. Interval-valued fuzzy sets, also called 2-dimensional fuzzy sets are more suitable than ordinary fuzzy sets (1- dimenional fuzzy sets) in mathematical modeling for uncertainties. Cu- bic sets (3-dimensinal fuzzy sets) are also an important extension of 1- dimensional fuzzy sets. It is also important to note that the (fuzzy) ideals have an essential role in the study of algebraic structures. In this pa- per we introduce cubic bi-ideals of semigroups and investigate interesting characterization theorems of these classes in terms of cubic bi-ideals. We introduce and study prime, strongly prime, semiprime, irreducible and strongly irreducible cubic bi-ideals of semigroups and characterize semi- groups in terms of semiprime and strongly prime cubic bi-ideals. We study the semigroups in which each cubic bi-ideal is prime. 2010 AMS Classification: 20N25, 03E72 Keywords: Semigroups, Cubic bi-ideal, Prime cubic bi-ideal, Strongly prime cubic bi-ideal, Semiprime cubic bi-ideal, Irreducible cubic bi-ideal, Strongly irreducible cubic bi-ideal. Corresponding Author: Asghar Khan (azhar4set@yahoo.com) 1. Introduction Fuzzy sets are initiated by Zadeh [17]. Since then fuzzy set theory developed by different algebraists and others has created great interest among researchers working in different branches of mathematics. Using this concept, Rosenfeld laid the foun- dation of fuzzy groupoids/fuzzy ideals [14]. In [10, 11, 12, 13], Kuroki introduced and studied the concepts of fuzzy ideal, fuzzy semiprime ideal and fuzzy bi-ideals in semigroups and characterized different classes of semigroups using these concepts. In [15, 16] Shabir et al. introduced the notions of prime fuzzy ideal and prime fuzzy bi-ideal of semigroups and discussed the class of those semigroups in which every