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Fundam. Prikl. Mat., 2007, Volume 13, Issue 4, Pages 121–144 (Mi fpm1066)  

This article is cited in 2 scientific papers (total in 2 papers)

Idempotent matrix lattices over distributive lattices

V. G. Kumarov

Murmansk State Pedagogical University

Abstract: 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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English version:
Journal of Mathematical Sciences (New York), 2008, 155:6, 877–893

Bibliographic databases:

UDC: 512.643

Citation: V. G. Kumarov, “Idempotent matrix lattices over distributive lattices”, Fundam. Prikl. Mat., 13:4 (2007), 121–144; J. Math. Sci., 155:6 (2008), 877–893

Citation in format AMSBIB
\Bibitem{Kum07}
\by V.~G.~Kumarov
\paper Idempotent matrix lattices over distributive lattices
\jour Fundam. Prikl. Mat.
\yr 2007
\vol 13
\issue 4
\pages 121--144
\mathnet{http://mi.mathnet.ru/fpm1066}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2366240}
\zmath{https://zbmath.org/?q=an:1157.06004}
\transl
\jour J. Math. Sci.
\yr 2008
\vol 155
\issue 6
\pages 877--893
\crossref{https://doi.org/10.1007/s10958-008-9248-3}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-57349138269}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. B. Poplavskii, “Ob idempotentakh algebry bulevykh matrits”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 12:2 (2012), 26–33  mathnet
    2. A. V. Zhuklina, “Sravnimye po stolbtsam i proportsionalnye po stolbtsam matritsy nad reshetkami”, Vladikavk. matem. zhurn., 14:3 (2012), 45–57  mathnet
  • Фундаментальная и прикладная математика
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