The invention of new sequences through classifying and counting fuzzy matrices
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The invention of new sequences through classifying and counting fuzzy matrices S. R. Kannan1 • Rajesh Kumar Mohapatra1
•
Tzung-Pei Hong2
Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The novelty of this paper is to construct several explicit formulas for the number of distinct fuzzy matrices of a finite order which leads us to invent new integer sequences and helps to develop fuzzy subgroups of some finite groups of matrices. In order to achieve the sequences, we analyze the behavioral study of a natural equivalence relation on the set of all fuzzy matrices of a given order. In addition, this paper derives some important relevant results by enumerating non-equivalent class of fuzzy matrices. We achieve these results by incorporating the notion of k-level fuzzy matrices, a-cuts and chains. Keywords Fuzzy matrices k-level fuzzy matrices a-cuts Chains of crisp matrices Flags Binomial numbers
1 Introduction The fuzzy matrices contribute a significant role in the development of artificial intelligence information systems, image processing, clustering analysis, medical diagnosis, decision making, and so on (Bellman and Zadeh 1970; Bezdek and Harris 1978; Kahraman 2007; Guleria and Bajaj 2019). But, the problem of counting all distinct fuzzy matrices of a finite order does not seem to have any attention so far. Classifying and counting the fuzzy matrices can successfully be used to establish the fuzzy subgroups (fuzzy normal subgroups) of some classes of finite groups of matrices: general linear group, permutation group, etc. In recent years, this subject has evolved remarkably in many areas and many interesting results have been proposed (Herrera et al. 2006; Ta˘rna˘uceanu 2009). Though the enumeration problem is quite a difficult job, many researchers have been still concentrating on both classifying and counting fuzzy subsets and fuzzy subgroups in the
Communicated by Kannan. & Rajesh Kumar Mohapatra [email protected] 1
Department of Mathematics, Pondicherry University (A Central University of India), Puducherry, India
2
Department of Computer Science and Information Engineering, National University of Kaohsiung, Kaohsiung, Taiwan
last few years (Murali and Makamba 2003a; Taˇrnaˇuceanu and Bentea 2008). Research works on counting the number of fuzzy subsets of a finite set were initially studied by Murali and Makamba (2003a). Later researchers such as Taˇrnaˇuceanu and Bentea (2008), Sˇesˇelja and Tepavcˇevic´ (2004), and Jain (2006) have taken the research to a further level. This work facilitates to an another direction to tackle the issue of counting fuzzy subgroups of finite groups. To derive the number of fuzzy subgroups of finite groups, researchers have firstly handled the special cases of finite abelian groups such as computing the number of distinct fuzzy subgroups of a square-free order finite cyclic group (Murali and Makamba 2003a), and cyclic group of order pn qm (p; q primes) (Murali and Makamba 2003a,
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