In this paper it is proved that the lattice of additive hereditary properties of finite graphs is completely distributive and that it does not satisfy the Jordan-Dedekind condition for infinite chains.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1048, author = {J\'an Jakub\'\i k}, title = {On the lattice of additive hereditary properties of finite graphs}, journal = {Discussiones Mathematicae - General Algebra and Applications}, volume = {22}, year = {2002}, pages = {73-86}, zbl = {1032.06003}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1048} }
Ján Jakubík. On the lattice of additive hereditary properties of finite graphs. Discussiones Mathematicae - General Algebra and Applications, Tome 22 (2002) pp. 73-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1048/
[000] [1] G. Birkhoff, Lattice Theory, (the 3-rd ed.), Amer. Math. Soc., Providence, RI, 1967.
[001] [2] M. Borowiecki, I. Broere, M. Frick, P. Mihók, and G. Semanisin, A survey of hereditary properties of graphs, Discussiones Math.- Graph Theory 17 (1997), 5-50. | Zbl 0902.05026
[002] [3] M. Borowiecki and P. Mihók, Hereditary properties of graphs, 'Advances in Graph Theory', Vishwa International Publications, Gulbarga 1991, 41-68.
[003] [4] G. Grätzer and E. T. Schmidt, On the Jordan-Dedekind chain condition, Acta Sci. Math. 18 (1957), 52-56. | Zbl 0079.04501
[004] [5] J. Jakubík, On the Jordan-Dedekind chain condition, Acta Sci. Math. 16 (1955), 266-269. | Zbl 0065.26602
[005] [6] J. Jakubík, A remark on the Jordan-Dedekind chain condition in Boolean algebras (Slovak), Casopis Pest. Mat. 82 (1957), 44-46.
[006] [7] J. Jakubík, On chains in Boolean algebras (Slovak), Mat. Fyz. Casopis SAV 8 (1958), 193-202. | Zbl 0086.25302
[007] [8] J. Jakubí k, Die Jordan-Dedekindsche Bedingung im direkten Produkt von geordneten Mengen, Acta Sci. Math. 24 (1963), 20-23. | Zbl 0118.02203
[008] [9] P. Mihók, On graphs critical with respect to generalized independence numbers, Colloq. Math. Soc. J. Bolyai 52 (1987), 417-421.
[009] [10] G.N. Raney, Completely distributive complete lattices, Proc. Amer. Math. Soc. 3 (1952), 677-680. | Zbl 0049.30304
[010] [11] G.N. Raney, A subdirect-union representation for completely distributive lattices, Proc. Amer. Math. Soc. 4 (1952), 518-522. | Zbl 0053.35201
[011] [12] G.N. Raney, Tight Galois connection and complete distributivity, Trans. Amer. Math. Soc. 97 (1960), 418-426. | Zbl 0098.02703
[012] [13] R. Sikorski, Boolean Algebras (the second edition), Springer-Verlag, Berlin 1964.
[013] [14] G. Szász, Generalization of a theorem of Birkhoff concerning maximal chains of a certain type of lattices, Acta Sci. Math. 16 (1955), 89-91. | Zbl 0064.02904