The structural analysis of fuzzy measures

Abstract

From quotient space based granular computing theory we explore the fuzzy measures of fuzzy sets. Based on the structural analysis of fuzzy sets, we present an "isotropism" assumption. Under this assumption we achieved the following insights: (1) the uniqueness of a fuzzy measure being isotropic on a finite complete semi-ordered set; (2) the necessary and sufficient condition of the isomorphism for fuzzy measure functions in fuzzy mathematics; (3) the necessary and sufficient condition that fuzzy measures have fuzzy and granular monotonicity; (4) under a certain assumption, the analytic expression for fuzzy measures. The results clarify the relationship between fuzzy measures and granular computing, reveal the essence of fuzzy measures, and provide a simple way of constructing fuzzy measures.

Overlapping community detection in complex networks using symmetric binary matrix factorization

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Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system

“…It is well known that the chaotic system has the characteristics of sensitivity of initial value and complexity of trajectory, so chaos has great potential application in many engineering fields, such as weak signal detection, [1][2][3][4] random number generator, [5,6] secure communication, [7][8][9][10][11][12][13] and image encryption. [14][15][16][17][18][19][20] Furthermore, Lin et al [21,22] studied the influence of electromagnetic radiation on the chaotic characteristics of neural networks, and found that appropriate electromagnetic radiation is helpful to treat some neurological diseases.…”

Hierarchical structures based on intuitionistic fuzzy tolerance relations

“…Just as a big umbrella it covers all the research on the theories, methodologies, technologies, and tools about granules, especially, under the concept of GrC, L. Zhang and B. Zhang [18][19][20][21][22][23][24] focus on the equivalence relationship among different grain-size worlds and on the problem that is mainly based on fuzzy sets, i.e., quotient space theory, fuzzy quotient space. In quotient space theory, hierarchical structures are expressed by the relationship on crisp sets in [20,24], and it is a core concept. The matrix is objective, operative, and can describe binary relations on crisp sets.…”