Yinyang bipolar fuzzy sets and fuzzy equilibrium relations: For clustering, optimization, and global regulation

Based on the notions of bipolar lattices and L-sets, YinYang bipolar fuzzy sets and fuzzy equilibrium relations are presented for bipolar clustering, optimization, and global regulation. While a bipolar L-set is defined as a bipolar equilibrium function L that maps a bipolar object set X over an arbitrary bipolar lattice B as L: X double right arrow B, this work focuses on the unit square lattice B-F = [-1, 0] x [0, 1]. A strong or weak bipolar fuzzy equilibrium relation in a bipolar set X is then defined as a reflexive, symmetric, and bipolar interactive (or transitive) fuzzy relation mu(R): X double right arrow B-F. Three types of bipolar alpha-level sets are presented for bipolar defuzzification and depolarization. It is shown that a fuzzy equilibrium relation is a non-linear bipolar generalization and/or fusion of multiple similarity relations, which induces disjoint or joint bipolar fuzzy subsets including quasi-coalition, conflict, and harmony sets. Equilibrium energy and stability analysis can then be utilized on different clusters for optimization and global regulation purposes. Thus, this work provides a unified approach to truth, fuzziness, and polarity and leads to a holistic theory for cognitive-map-based visualization, optimization, decision, global regulation, and coordination. Basic concepts are illustrated with a simulation in macroeconomics.