Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattice
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).
scientific article; zbMATH DE number 6127929
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