Fuzzy equivalence relation and its multigranulation spaces

Multigranulation rough sets focus on describing uncertain sets in different granularity spaces. Different approximation accuracy of an uncertain set can be obtained through changing knowledge granularities according to user’s need. However, it is a bottleneck for establishing a multigranulation space to better describe an uncertain set in a proper granulation space. And the typical way of a multigranulation space is a hierarchical quotient space determined by a fuzzy equivalence relation. In this paper, a new concept, granularity of binary relation (GB) is put forward firstly. And knowledge granulation (GK) is proved to be a special case of GB. The changing rules of granularity series for describing a hierarchical quotient space structure are proposed, and approximation accuracy series of an uncertain set is presented. For automatically searching optimal knowledge space according to user’s requirement, a utility function with both knowledge space granularity and approximation accuracy is proposed. Then classification isomorphism and granularity isomorphism between two hierarchical quotient space structures are defined and discussed. Through the hierarchical quotient space structure of a fuzzy equivalence relation, the essence of classification ability of a fuzzy equivalence relation is also revealed. Finally, a new algorithm for establishing isomorphic fuzzy equivalence relations based on a hierarchical quotient space structure is presented, and this algorithm can further explain the nature of those isomorphic fuzzy equivalence relations from a new perspective.