Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an $n$-element set $\operatorname{Qord}(n)$ is not graduated for $n\geq3$ and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval $([I,J]_\leq,\leq)$ of idempotent $(n\times n)$-matrices over $\{\tilde0,\tilde1\}$-lattices is isomorphic to the lattice of quasi-orders $\operatorname{Qord}(n)$. Using this isomorphism, we calculate the lattice height of idempotent $(\tilde0,\tilde1)$-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices.