FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(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 $Qord(n)$ 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 $([I,\; J]$_{£},
£)
of idempotent $(n$´ n)-matrices over $\$\; \backslash \{\backslash tilde\; 0,\; \backslash tilde\; 1\backslash \}\; \$$-lattices
is isomorphic to the lattice of
quasi-orders $Qord(n)$.
Using this isomorphism, we calculate the lattice height of idempotent
$\$\; (\backslash tilde\; 0,\; \backslash tilde\; 1)\; \$$-matrices.
We obtain a structural criterion of idempotent matrices over
distributive lattices.

Location: http://mech.math.msu.su/~fpm/eng/k07/k074/k07407h.htm

Last modified: November 28, 2007