On AFS algebra––Part I

Abstract

This paper addresses the properties and structures of EI algebra and EI n algebra in the framework of Liu [Fuzzy Sets and Systems 95 (1998) 179]. The main contribution of the paper is that the expressions of both Ã-irreducible elements (or^-irreducible elements) in molecular lattice ðEM; PÞ, and ðEX 1 Á Á Á X n M; PÞ are developed, and that the standard minimal family of elements in molecular lattice ðEM; PÞ and ðEX 1 Á Á Á X n M; PÞ is formulated. It is proved that neither molecular lattice ðEM; PÞ nor molecular lattice ðEX 1 Á Á Á X n M; PÞ is a fuzzy lattice.

A fuzzy-based customer clustering approach with hierarchical structure for logistics network optimization

“…According to Axiomatic Fuzzy Set (AFS) theory (Liu, 1998a(Liu, , 1998bLiu, Wang, & Chai, 2005;Zhang, Liang, & Tong, 2004a, Zhang, Liang, & Tong, 2004b, (M,s,X) is called an Axiomatic Fuzzy Set (AFS) structure, X is called a sample set, M is called an attribute (concept) set, s is called structure; Let R be the binary relation of the sample set X, and the binary relation contains sub-preference relations and preference relations, and the corresponding concept of sub-preference relation is called simple concept, and the preference relation is called complex concept. Definition 4.…”

“…The next step is to conduct clustering analysis for dividing the customers into different clusters. Axiomatic fuzzy set (AFS) theory logic has been proven as an effective approach for tackling human perception related clustering problem (Liu, 1998a;Zhang et al, 2004aZhang et al, , 2004b. Since the traditional AFS method is not appropriate to include a large number of criteria (Liu et al, 2005), an improved AFS theory logic algorithm is introduced as follows:…”

Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory I

“…AFS theory is based on AFS structures, which is a special kind of combinatorial objects [3], and the AFS algebras, which is a family of completely distributive lattices. For detailed mathematical properties of the AFS algebras, see [7,11,23,24].…”

Concept analysis via rough set and AFS algebra

“…Recently, AFS algebra has been further developed and applied to fuzzy clustering analysis [13,19], fuzzy decision trees [12] and concept representations [9], etc. About the detail properties of AFS algebra, please see [9][10][11]33].…”