Finitely generated lattices with completely modular elements among generators
Algebra i logika, Tome 52 (2013) no. 6, pp. 657-666.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
@article{AL_2013_52_6_a0,
     author = {A. G. Gein and M. P. Shushpanov},
     title = {Finitely generated lattices with completely modular elements among generators},
     journal = {Algebra i logika},
     pages = {657--666},
     publisher = {mathdoc},
     volume = {52},
     number = {6},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2013_52_6_a0/}
}
TY  - JOUR
AU  - A. G. Gein
AU  - M. P. Shushpanov
TI  - Finitely generated lattices with completely modular elements among generators
JO  - Algebra i logika
PY  - 2013
SP  - 657
EP  - 666
VL  - 52
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2013_52_6_a0/
LA  - ru
ID  - AL_2013_52_6_a0
ER  - 
%0 Journal Article
%A A. G. Gein
%A M. P. Shushpanov
%T Finitely generated lattices with completely modular elements among generators
%J Algebra i logika
%D 2013
%P 657-666
%V 52
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2013_52_6_a0/
%G ru
%F AL_2013_52_6_a0
A. G. Gein; M. P. Shushpanov. Finitely generated lattices with completely modular elements among generators. Algebra i logika, Tome 52 (2013) no. 6, pp. 657-666. http://geodesic.mathdoc.fr/item/AL_2013_52_6_a0/

[1] M. Stern, Semimodular lattices. Theory and applications, Encyclopedia Math. Appl., 73, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[2] S. Maeda, “On distributive pairs in lattices”, Acta Math. Hung., 45:1/2 (1985), 133–140 | DOI | MR | Zbl

[3] L. R. Wilcox, “Modularity in the theory of lattices”, Ann. Math. (2), 40 (1939), 490–505 | DOI | MR | Zbl

[4] S. Maeda, “On finite-modular atomistic lattices”, Algebra Univers., 12 (1981), 76–80 | DOI | MR | Zbl

[5] B. M. Vernikov, M. V. Volkov, “Modular elements of the lattice of semigroup varieties. II”, 70. Arbeitstagung Allgemeine Algebra, Proc. of the 70th workshop on general algebra (Vienna, Austria, May 26–29, 2005), Contrib. General Algebra, 17, eds. G. Dorfer et al., Verlag Johannes Heyn, Klagenfurt, 2006, 173–190 | MR | Zbl

[6] B. M. Vernikov, “Verkhnemodulyarnye elementy reshëtki mnogoobrazii polugrupp. II”, Fundam. prikl. matem., 14:7 (2008), 43–51 | MR

[7] R. Schmidt, “Modulare Untergruppen endlicher Gruppen”, Ill. J. Math., 13 (1969), 358–377 | MR | Zbl

[8] R. K. Amayo, J. Schwarz, “Modularity in Lie algebras”, Hiroshima Math. J., 10 (1980), 311–322 | MR | Zbl

[9] G. Birkhoff, “Neutral elements in general lattices”, Bull. Am. Math. Soc., 46 (1940), 702–705 | DOI | MR | Zbl

[10] A. G. Gein, M. P. Shushpanov, “O podreshëtke, porozhdënnoi modulyarnymi elementami”, Algebra i lineinaya optimizatsiya, Tez. dokl. Mezhd. algebr. konf., posvyasch. 100-letiyu S. N. Chernikova (Ekaterinburg, 14–19 maya 2012 g.), Izd-vo UMTs-UPI, Ekaterinburg, 2012, 47

[11] A. G. Gein, M. P. Shushpanov, “O reshëtkakh, porozhdënnykh vpolne modulyarnymi elementami”, Maltsevskie chteniya, Tez. dokl. Mezhd. konf. (Novosibirsk, 13–15 noyabrya 2012 g.), IM SO RAN, Novosibirsk, 2012, 52

[12] A. G. Gein, M. P. Shushpanov, “Ob opredelyayuschikh sootnosheniyakh svobodnoi modulyarnoi reshëtki ranga 3”, Izv. vuzov. Matem., 2013, no. 10, 69–72 | Zbl