The determinant of square intuitionistic fuzzy matrices (Q2765944)

The authors introduce the concept of an intuitionistic fuzzy matrix (ifm) and study properties of ifm's. An ifm \({\mathcal A}\) is a matrix of pairs \({\mathcal A}=[(a_{ij}, a'_{ij})]\) of nonnegative reals satisfying \(a_{ij}+a'_{ij}\leq 1\) for all \(i,j.\) The matrix operations \({\mathcal A}+{\mathcal B}=[(a_{ij}\vee b_{ij}, a'_{ij}\wedge b'_{ij})],\) \({\mathcal A}{\mathcal B}=[(\bigvee_k a_{ik}\vee b_{kj}, \bigwedge_k a'_{ik}\wedge b'_{kj})]\) are defined given ifm's \({\mathcal A},{\mathcal B}\) of respectively compatible sizes. The determinant is \(|{\mathcal A}|= [\bigvee_{\sigma \in S_n} a_{1\sigma(1)}\wedge \cdots \wedge a_{n\sigma(n)}, \bigwedge_{\sigma \in S_n} a'_{1\sigma(1)}\vee \cdots \vee a'_{n\sigma(n)})].\) Classical fuzzy matrix theory, see e.g. \textit{J. B. Kim, A. Baartmans} and \textit{N. S. Sahadin} [Fuzzy Sets Syst. 29, No. 3, 349-356 (1989; Zbl 0668.15004)] is obtained from ifm-theory via the correspondence \([(a_{ij})] \leftrightarrow [(a_{ij},1-a_{ij})]\).