(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 4, PAGES 121-144

## Idempotent matrix lattices over distributive lattices

V. G. Kumarov

Abstract

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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 $Qord\left(n\right)$ is not graduated for $n$³ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval $\left(\left[I, J\right]$£, £) of idempotent $\left(n$´ n)-matrices over $\\left\{\tilde 0, \tilde 1\\right\}$-lattices is isomorphic to the lattice of quasi-orders $Qord\left(n\right)$. Using this isomorphism, we calculate the lattice height of idempotent $\left(\tilde 0, \tilde 1\right)$-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices.

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