Determinant theory for \(D_{01}\)-lattice matrices (Q1896328)

The paper deals with permanents of matrices [cf. \textit{H. Minc}, Permanents (1978; Zbl 0401.15005)]. For a matrix \(A\) over a bounded distributive lattice \((L, \vee, \wedge, 0,1)\) it is proved that: \[ \begin{aligned} \text{per} (A) \wedge \text{per} (B) & \leq \text{per} (AB) \leq \text{per} (A \vee B), \\ \text{per} \bigl( A \text{adj} (A) \bigr) & = \text{per} (A) = \text{per} \bigl( \text{adj} (A)A \bigr). \end{aligned} \] The author also proves a lattice version of the Cayley-Hamilton Theorem suggested by \textit{J. B. Kim}, \textit{A. Baartmans} and \textit{N. S. Sahadin} [Fuzzy Sets Syst. 29, No. 3, 349-356 (1989; Zbl 0668.15004)].