Equilibrium relations and bipolar fuzzy clustering

Equilibrium relations bipolar fuzzy sets are introduced. Equilibrium relations are bipolar reflexive, symmetrical, and transitive. It is proved that: (1) crisp equilibrium relations induce hard partitions of harmony sets, coalition sets, and conflict sets; and (2) fuzzy equilibrium relation induces fuzzy harmony, coalition, and conflict sets which can be either disjoint or overlapped. Conditions for disjoint bipolar fuzzy partitioning are established with fuzzy equilibrium relations. It is shown that crisp equilibrium relations and fuzzy equilibrium relations are natural generalizations of classical equivalence relations and fuzzy similarity relations, respectively. It is illustrated that equilibrium relations can be used for bipolar clustering