Geodesic
Idempotent matrix lattices over distributive lattices
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 121-144.

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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. 
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     url = {http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a6/}
}
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V. G. Kumarov. Idempotent matrix lattices over distributive lattices. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 121-144. http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a6/

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