On the min-max composition of fuzzy matrices
The paper examines the min-max product \[ (A*B)_{ij} = \min_{1\leq k\leq n} \max (a_{ik}, b_{kj}), \quad i=1, \dots, m,\;j=1, \dots,p \] of matrices \(A,B\) over the semiring \(([0,1], \min, \max)\) [cf. e.g. Chapter 3 in: \textit{A. Kandel}, \textit{S. C. Lee}, Fuzzy switching and automata, Crane Russak, New York 1979; Zbl 0406.94022)]. It is proved that \(I^*(R^*R^{-1})\) is an idempotent relation (under the max-min composition, \(I\) -- identity relation). Other properties of the min-max product are a consequence of results on the fuzzy relation composition \(A\circ B\) in virtue of Proposition 3.12: Min-max and max-min products are dual in de Morgan laws and \(A^*B= (A^c \circ B^c)^c\).
scientific article; zbMATH DE number 919432