Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattice
type: Item
instance of: scholarly article
title: Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattice
zbMATH Open document ID: 1271.15015
author: Amal Kumar Adak
author: Monoranjan Bhowmik
author: Madhumangal Pal
published in: Fuzzy Information and Engineering
publication date: 2013-01-22T00:00:00Z
review text: Generalized intuitionistic fuzzy sets were introduced by \textit{T. K. Mondal} and \textit{S. K. Samanta} [J. Fuzzy Math. 10, No. 4, 839-861 (2002; Zbl 1028.03048)]. A base of this generalization is the lattice \(L= [0,1]\times [0,0.5] \cup [0,0.5]\times [0,12]\) with operations \((a,b)\vee (c,d) = (\max(a,c),\min(b,d))\), \((a,b)\wedge (c,d) = (\min(a,c),\max(b,d))\) (because of mixed order we have \(\min L = (0,1)\), \(\max L = (1,0)\)). The paper deals with matrices over this distributive lattice, called here generalized intuitionistic fuzzy matrices. The authors consider the permanent and the adjoint of such matrices and some properties of nilpotent matrices. Reviewer's remark: The list of references is used mainly in the introduction without references of the presented results. In particular, on pp. 374--378, some properties of posets and lattices are reproved, e.g. Theorems 2.1 and 3.4 are from the book by \textit{G. Birkhoff} [Lattice theory. Third (new) ed. Providence, R.I.: American Mathematical Society (1967; Zbl 0153.02501)] (cf. Theorem 3 on p. 4 and Theorem 10 on p. 12). Many of the other results are based on the paper by \textit{Y.-J. Tan} [Fuzzy Sets Syst. 151, No. 2, 421--433 (2005; Zbl 1062.06021)], e.g. Propositions 5.1, 5.3, 5.5 and Theorems 6.1--6.3 are particular cases of Lemma 2.4, Propositions 3.2, 3.4 and Theorems 4.1--4.3 from the above paper, respectively (because \(L\) is a particular case of distributive lattice).
Mathematics Subject Classification ID: 15B15
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 08A72
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 6127929
zbMATH Keywords: distributive lattice
zbMATH Keywords: matrices over lattice
zbMATH Keywords: intuitionistic matrix
zbMATH Keywords: nilpotent matrix
zbMATH Keywords: irreflexive matrix
zbMATH Keywords: transitive matrix
zbMATH Keywords: permanent
zbMATH Keywords: adjoint
zbMATH Keywords: fuzzy matrix
reviewed by: Józef Drewniak
MaRDI profile type: Publication
full work available at URL: s12543-012-0121-1
OpenAlex ID: W2089072004
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DOI: 10.1007/S12543-012-0121-1
description: scientific article; zbMATH DE number 6127929