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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2013, Volume 52, Number 6, Pages 657–666 (Mi al612)

Finitely generated lattices with completely modular elements among generators

A. G. Gein, M. P. Shushpanov

El'tsin Ural Federal University, ul. Mira 19, Yekaterinburg, 620002, Russia

Abstract: We look at the concept of a completely modular element of a lattice, which is the modular analog of the well-known concept of a neutral element of a lattice. It is proved that a lattice is modular if it is generated by three elements of which two are completely modular. Also it is shown that a lattice generated by $n$, $n>3$, completely modular elements must not necessarily be modular.

Keywords: modular lattices, free lattices, modular elements.

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English version:
Algebra and Logic, 2014, 52:6, 435–441

Bibliographic databases:

UDC: 512.565
Revised: 28.11.2013

Citation: A. G. Gein, M. P. Shushpanov, “Finitely generated lattices with completely modular elements among generators”, Algebra Logika, 52:6 (2013), 657–666; Algebra and Logic, 52:6 (2014), 435–441

Citation in format AMSBIB
\Bibitem{GeiShu13} \by A.~G.~Gein, M.~P.~Shushpanov \paper Finitely generated lattices with completely modular elements among generators \jour Algebra Logika \yr 2013 \vol 52 \issue 6 \pages 657--666 \mathnet{http://mi.mathnet.ru/al612} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3242615} \transl \jour Algebra and Logic \yr 2014 \vol 52 \issue 6 \pages 435--441 \crossref{https://doi.org/10.1007/s10469-014-9258-9} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000332578600001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84897632516} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. G. Geǐn, M. P. Shushpanov, “Sufficient conditions for the modularity of the lattice generated by elements with properties of modular type”, Siberian Math. J., 56:4 (2015), 631–636
2. A. G. Gein, “Finitely generated lattices with $M$-standard elements among generators”, Russian Math. (Iz. VUZ), 60:3 (2016), 14–17
3. M. P. Shushpanov, “O beskonechnosti svobodnoi $3$-porozhdennoi reshetki s odnim levomodulyarnym porozhdayuschim”, Sib. elektron. matem. izv., 14 (2017), 528–532
4. A. G. Gein, M. P. Shushpanov, “Modularity and distributivity of $3$-generated lattices with special elements among generators”, Algebra and Logic, 56:1 (2017), 1–12
5. M. P. Shushpanov, “On $3$-generated lattices with a completely modular element among generators”, Algebr. Universalis, 78:3 (2017), 377–387
6. A. G. Gein, M. P. Shushpanov, “Free $3$ -generated lattices with two semi-normal generators”, Order-J. Theory Ordered Sets Appl., 35:2 (2018), 247–252
7. M. P. Shushpanov, “Finiteness of a $3$-generated lattice with seminormal and coseminormal elements among generators”, Algebra and Logic, 57:3 (2018), 237–247
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