YinYang bipolar lattices and L-sets for bipolar knowledge fusion, visualization, and decision

Yin Yang bipolar sets, bipolar lattice, bipolar L-crisp sets, and Bipolar L-fuzzy sets are presented for bipolar information/knowledge fusion, visualization, and decision. First, a bipolar lattice B is defined as a 4-tuple (B, circle plus, &, circle times) in which every pair of elements has a bipolar lub (blub circle plus), a bipolar glb (bglb &), and a cross-pole glb (cglb circle times). A bipolar L-set (crisp or fuzzy) B = (B-, B+) in X to a bipolar lattice B-L is defined as a bipolar equilibrium function or mapping B : X double right arrow B-L. A strict bipolar lattice B is defined as a 7-tuple (B, equivalent to circle plus, circle times, &, -, (SIC) double right arrow) that delegates a class of isomorphic bounded and complemented bipolar lattices. A refined and generalized 9-set axiomatization is presented on a class of strict bipolar lattices. The notions of bipolar L-relations and equilibrium relations are introduced as bipolar L-sets. Remarkably, YinYang bipolar L-sets lead to a bipolar universal modus ponens (BUMP) which presents a unified nonlinear bipolar generalization of classical modus ponens and builds a bridge from a linear, static, and closed world to a non-linear, dynamic, and open world of equilibria, quasiequilibria, and non-equilibria for bipolar information/knowledge fusion, visualization, and decision. A number of potential applications are outlined.