Further Research On Transitive Closure And Interior Of A Fuzzy Relation

In order to apply fuzzy relations in various fields, one introduces manyproperties of fuzzy relations, among which transitivity is the most im-portant and extensively used one. For example, transitivity is necessaryin the study of fuzzy clustering analysis, fuzzy choice function, rankingfuzzy quantities and fuzzy preference modeling theory etc.. However, thosedata obtained from real world hardly satisfy the transitivity, or any othertransitivity-related property. Consequently, we have to modify the obtainedrelation so that the desired property is satisfied. To achieve the goal, it iscommon to find the closure or interior of the given relation.Considering that, as far as some certain property is concerned, theclosure or interior of a relation does not necessarily exist, one often turnsto seek a maximum interior or minimum closure of the relation in question.In this paper, we investigate these two concepts in detail and present someconstructive methods of the maximum interior or the minimum closure ofthe relation. The main work is summarized as follows:Firstly, using the concept introduced by Defays and based on the method of finding a maximum T-transitive (min-transitive) interior of afuzzy relation on a finite universe proposed by Fodor etc., we suggest sev-eral alternative constructing approaches, and prove that the fuzzy relationsobtained through our approaches are all maximum T-transitive interiorswhen T is a left-continuous t-norm. As a result, the study of the maximumtransitive interior is largely enriched. Here in the following are the details:(1) Compared with the existing method to compute maximum transitiveinterior (?) of a fuzzy relation, we use the last row of R as the last rowof maximum transitive interior. Then compute the rows of (?) backwardsrespectively the second last row, the third last row,…, until all the rowsof (?) are figured out.(2) In order to get more maximum T-transitive interiors of a fuzzyrelation R, we also give a medial row method of a fuzzy relation with oddnumber order, in which the medial row of R is chosen as the same row of(?). Then the remaining rows are defined respectively. Another maximumT-transitive interior can be found.(3) Furthermore, we present several column constructing method, inwhich the first column, the last column, and the medial column of R arechosen as the same column of (?) respectively, others are defined by a similarmethod to (2). In doing so, different maximum T-transitive interiors maybe generated.Next, S-negatively transitive closure and interior are investigated. Af-ter the definitions of S-negatively transitive interior and the minimum S-negatively transitive closure are given, their properties are discussed, andsome simplified algorithms are put forth in some particular cases. Mean- while, we introduce the concept of coimplication to construct the minimumS-negatively transitive closure directly.In a summary, in the thesis, we present several different methods toconstruct maximum interiors and minimum closures of a fuzzy relationon a finite universe. In order to obtain many maximum interiors andmany minimum closures through our approaches rapidly, we program thealgorithms in C language and realize them on computer. The research inour work enriches the study of transitivity-related properties and lays afoundation for finding all maximum interiors and minimum closures in thefuture.