Three-way fuzzy concept lattice representation using neutrosophic set
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ORIGINAL ARTICLE
Three-way fuzzy concept lattice representation using neutrosophic set Prem Kumar Singh1
Received: 23 June 2016 / Accepted: 10 August 2016 Springer-Verlag Berlin Heidelberg 2016
Abstract Recently, three-way concept lattice is studied to handle the uncertainty and incompleteness in the given attribute set based on acceptation, rejection, and uncertain regions. This paper aimed at analyzing the uncertainty and incompleteness in the given fuzzy attribute set characterized by truth-membership, indeterminacy-membership, and falsity membership functions of a defined single-valued neutrosophic set. For this purpose a method is proposed to generate the component wise three-way formal fuzzy concept and their hierarchical order visualization in the fuzzy concept lattice using the properties of neutrosophic graph, neutrosophic lattice, and Go¨del residuated lattice with an illustrative example. Keywords Formal concept analysis Fuzzy concept lattice Formal fuzzy concept Three-way concept lattice Neutrosophic set
1 Introduction In the early eighties a mathematical model called as Formal Concept Analysis (FCA) was introduced by wille [1] based on applied lattice theory for knowledge processing tasks. In the last decade the properties of FCA has been applied in various research fields [2]. FCA provides some set of patterns called as formal concepts from a data set in form of binary matrix—(X, Y, R) where row represents set of objects (X), column represents set of attributes (Y), and & Prem Kumar Singh [email protected]; [email protected] 1
Amity Institute of Information Technology, Amity University, Sector-125, Noida 201313, UP, India
each entries in the matrix represents binary relations among them (R X Y). Further its provides hierarchical order visualization of generated formal concepts through a defined concept lattice structure [3]. For precise representation of uncertainty and incompleteness the mathematics of FCA is augmented with fuzzy context [4], heterogeneous context [5], interval-valued fuzzy context [6], bipolar fuzzy context [7], linked fuzzy context [8], possibilitytheoretic [9], and rough set [10] based formal context. Recently these extensions of FCA and their future trends are analysed based on their suitable applicability in a defined universe [2]. Recently the research trends to analysis of three-way formal concept analysis [11], its connection with classical concept lattice [12], multi-scaled concept lattice [13], triadic-decision context [14], threeway incomplete context [15, 16], three-way cognitive concept learning [17] at different granulation [18]. Further three-way decision space [19], with fuzzy sets [20], hesitant fuzzy sets [21] and their knowledge reduction [22] is also studied. Motivated from these recent studied current paper focused on rigorous analysis of three-way fuzzy concept lattice based on truth-membership function, indeterminacy-membership function, and falsity-membership function of a defined neutrosophic set. Fuzzy concept lattice repres
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