Bipolar Fuzzy Sets and Equilibrium Relations

Introduction

Based on the concepts of set, binary relation, equivalence relation, fuzzy set,fuzzy relation, similarity relation, we introduce the concepts of bipolar fuzzy relation, fuzzy equilibrium relation, bipolar partitioning, and bipolar clustering in this chapter. Decision analysis in agent-oriented macroeconomics is illustrated using equilibrium-based bipolar clustering.

While set theory provides the doctrine for classical mathematical abstraction, Zadeh’s fuzzy set theory (Zadeh 1965) presents a serious challenge to the doctrine. The fuzzy set research community successfully fought the decisive battles in seeking recognition in the last quarter century before the new millennium with a number of heated debates and dramatic stories. Now fuzzy set theory has been widely accepted and is commonly considered an established field in both academic and applied terms.

It can be observed that fuzzification in Zadeh’s fuzzy set theory preserves the truth-based property of classical set theory with the addition of an infinite number of truth gray levels which can be defuzzified to true or false (1 or 0). Therefore, fuzzy sets can be deemed an extension of classical set theory that is based largely on Aristotle’s philosophy of science where the universe consists of a set of truth objects. Fundamentally, the fuzzy extension is realized by fuzzifying the bivalent lattice {0,1} to the fuzzy lattice [0,1].

As stated in Postulate 1.1 (Ch. 1), truth and fuzziness cannot serve as the most fundamental property of nature because they are not physical but logical properties subjected to the LAFIP paradox. Three decades after inventing his fuzzy set theory, the founder of fuzzy sets Lotfi Zadeh concluded that “Causality is Undefinable” (Zadeh, 2001). Evidently, the purpose of fuzzy set theory has not been for providing logically definable causality or providing the logical foundation for physics as envisioned by Einstein in his quote: “For the time being we have to admit that we do not possess any general theoretical basis for physics which can be regarded as its logical foundation.”

Although it conveys no physical information if we say that the universe is fuzzy, it does make sense if we say the universe is a quasi- or fuzzy equilibrium. Thus, based on bipolar sets, bipolar dynamic logic (BDL), and bipolar quantum lattice as introduced in earlier chapters, we discuss bipolar fuzzy sets and fuzzy equilibrium relations in this chapter. The word “bipolar fuzzy sets” was first coined in 1994 (Zhang, 1994). Further development of the theory is documented in (Zhang, 1996, 1998, 2003b; Zhang & Zhang, 2004). The formal presentation of the theory is documented in (Zhang, 2005a, 2006). (The author acknowledges Lotfi Zadeh (2006) for publically recognizing bipolar fuzzy sets.)

From Chapters 3 – 4, it is clear that:

• 1.

  The bipolar crisp lattice B₁ = {-1,0}× {0,+1} (Ch. 3) is the corner lattice of B_F = [-1,0]×[0,+1];

• 2.

  B₁ is a polarization of the bivalent lattice {0,1};

• 3.

  B_F is a polarization of the fuzzy lattice [0,1]; and

• 4.

  Both B₁ and B_F are strict bipolar lattices (Ch. 4) (Zhang, 2005a).

In this chapter, we use the terms bipolar crisp sets and bipolar fuzzy sets to distinguish their mapping to B₁ and B_F, respectively, and we follow the basic concepts and notations of bipolar quantum lattice, bipolar sets, bipolar relation, bipolar transitivity, and bipolar equilibrium relation as defined in Chapters 3-4.