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Idempotent matrix lattices over distributive lattices. (English. Russian original) Zbl 1157.06004

J. Math. Sci., New York 155, No. 6, 877-893 (2008); translation from Fundam. Prikl. Mat. 13, No. 4, 121-144 (2007).
Summary: 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 \(\text{Qord}(n)\) is not graduated for \(n \geq 3\) 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 \(\{\tilde 0,\tilde 1\}\)-lattices is isomorphic to the lattice of quasi-orders \(\text{Qord}(n)\). Using this isomorphism, we calculate the lattice height of idempotent \((\tilde 0,\tilde 1)\)-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices.

MSC:

06B05 Structure theory of lattices
15A30 Algebraic systems of matrices
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