Geodesic
A generalization of the Birkhoff–Whitney theorem for hereditary systems
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 4, pp. 66-75
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The notion of hereditary system is a natural generalization of the notion of matroid. We prove the following generalization of the Birkhoff–Whitney theorem for hereditary systems: the lattice of closed sets of any hereditary system does not contain infinite chains and, vice versa, any nonempty lattice without infinite chains is isomorphic to the lattice of closed sets of some hereditary system. In particular, any finite nonempty lattice is isomorphic to the lattice of closed sets of some finite hereditary system.
Keywords: hereditary system, matroid, closure operator, geometric lattice, hypergraph. 
@article{TIMM_2011_17_4_a5,
     author = {M. Yu. Vyplov},
     title = {A generalization of the {Birkhoff{\textendash}Whitney} theorem for hereditary systems},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {66--75},
     year = {2011},
     volume = {17},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a5/}
}
TY  - JOUR
AU  - M. Yu. Vyplov
TI  - A generalization of the Birkhoff–Whitney theorem for hereditary systems
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2011
SP  - 66
EP  - 75
VL  - 17
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a5/
LA  - ru
ID  - TIMM_2011_17_4_a5
ER  - 
%0 Journal Article
%A M. Yu. Vyplov
%T A generalization of the Birkhoff–Whitney theorem for hereditary systems
%J Trudy Instituta matematiki i mehaniki
%D 2011
%P 66-75
%V 17
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a5/
%G ru
%F TIMM_2011_17_4_a5
M. Yu. Vyplov. A generalization of the Birkhoff–Whitney theorem for hereditary systems. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 4, pp. 66-75. http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a5/

[1] Aigner M., Kombinatornaya teoriya, 1982, Mir, M., 558 pp. | MR

[2] Birkgof G., Teoriya reshetok, Nauka, M., 1984, 568 pp. | MR

[3] Vyplov M. Yu., Ilev V. P., “O gipergrafakh, sootvetstvuyuschikh operatoram slabogo zamykaniya”, Diskretnye modeli v teorii upravlyayuschikh sistem, URL: , 2009, 30–34 http://www.dmconf.ru/dm8/proceedings.pdf

[4] Crapo H. H., Rota G.-C., On the foundations of combinatorial theory, v. II, Combinatorial geometries, MIT Press, Cambridge, Mass., 1970, 293 pp. | MR

[5] Grëotshel M., Lovász L., “Combinatorial optimization”, Handbook of Combinatorics, v. 2, eds. R. O. Graham, M. Grtshel, L. Lovász, Elsevier Science B.V., Amsterdam, 1995, 1341–1598 | MR

[6] Whitney H., “On the abstract properties of linear dependence”, Amer. J. Math., 57:3 (1935), 509–533 | DOI | MR