AFS Topology and Its Applications

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In this chapter, first we construct some topologies on the AFS structures, discuss the topological molecular lattice structures on EI, * EI, EII, * EII algebras, and elaborate on the main relations between these topological structures. Second, we apply the topology derived by a family of fuzzy concepts in EM, where M is a set of simple concepts, to analyze the relations among the fuzzy concepts. Thirdly, we propose the differential degrees and fuzzy similarity relations based on the topological molecular lattices generated by the fuzzy concepts on some features. Furthermore, the fuzzy clustering problems are explored using the proposed differential degrees and fuzzy similarity relations. Compared with other fuzzy clustering algorithms such as the Fuzzy C-Means and k-nearest-neighbor fuzzy clustering algorithms, the proposed fuzzy clustering algorithm can be applied to data sets with mixed feature variables such as numeric, Boolean, linguistic rating scale, sub-preference relations, and even descriptors associated with human intuition. Finally, some illustrative examples show that the proposed differential degrees are very effective in pattern recognition problems whose data sets do not form a subset of a metric space such as the Eculidean one. This approach offers a promising avenue that could be helpful in understanding mechanisms of human recognition.