Bipolar logic and bipolar fuzzy partial ordering for clustering and coordination

A bipolar crisp logic and a bipolar fuzzy logic are introduced that generalize Boolean logic and Zadeh's fuzzy logic, respectively, to the strict bipolar spaces {-1,0} x {0,1} and [-1,0] x [0,1] Based on the bipolar fuzzy logic,fuzzy relations, fuzzy equilibrium relations, fuzzy equilibrium classes, and bipolar partial ordering are introduced, which are generalizations of crisp and. fuzzy binary relations, equivalence/similarity relations, equivalence/similarity. classes, and classical unipolar partial ordering. Energy and stability of fuzzy equilibrium relations are discussed. Examples are presented to illustrate basic concepts for bipolar clustering and coordination.