FACTORIZATION OF INTUITIONISTIC FUZZY PREFERENCE RELATIONS JOHN N. MORDESON Department of Mathematics Creighton University, Omaha, Nebraska 68178, USA mordes@creighton.edu TERRY D. CLARK Department of Political Science Creighton University, Omaha, Nebraska, USA tclark@creighton.edu KAREN ALBERT Department of Political Science, University of Nebraska Lincoln, Nebraska 68588, USA karenalbert@huskers.unl.edu The proofs of many factorization results for an intuitionistic fuzzy binary relation h  ;  i involve dual proofs, one for   with respect to a t-conorm  and one for   with respect to a t-norm . In this paper, we show that one proof can be obtained from the other by considering  and  dual under an involutive fuzzy complement. We provide a series of singular proofs for commonly de¯ned norms and conorms. Keywords: Intuitionistic fuzzy preference relation; factorization; t-conorm; t-norm; involutive fuzzy complement; asymmetric fuzzy preference relation; duality. 1. Introduction The factorization of a weak preference relation into a strict preference relation and an indi®erence relation is unique in the crisp case. In the fuzzy case, there may be several factorizations of a fuzzy weak preference relation. Let X be a non-empty set. A function from X into the closed interval ½0; 1 is called a fuzzy subset of X 1 . Let  be a fuzzy subset of X. Then the support of , SuppðÞ, is de¯ned to be fx 2 XjðxÞ > 0g. The cosupport of , CosuppðÞ, is de¯ned to be fx 2 XjðxÞ < 1g. Let t 2½0; 1. Then the t-level set of , written  t , is de¯ned to be the set fx 2 XjðxÞ tg. A fuzzy binary relation on X is a fuzzy subset of the set of ordered pairs X  X. We let FRðXÞ denote the set of all fuzzy binary relations on X. Let ;  be fuzzy subsets of X. We write    if ðxÞ ðxÞ for all x 2 X. We write New Mathematics and Natural Computation Vol. 10, No. 1 (2014) 1–25 # . c World Scienti¯c Publishing Company DOI: 10.1142/S179300571450001X 1