Geodesic
Direct product decompositions of infinitely distributive lattices
Mathematica Bohemica, Tome 125 (2000) no. 3, pp. 341-354

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
Let $\alpha$ be an infinite cardinal. Let $\Cal T_\alpha$ be the class of all lattices which are conditionally $\alpha$-complete and infinitely distributive. We denote by $\Cal{T}_\sigma'$ the class of all lattices $X$ such that $X$ is infinitely distributive, $\sigma$-complete and has the least element. In this paper we deal with direct factors of lattices belonging to $\Cal T_\alpha$. As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class $\Cal T_\sigma'$.
Let $\alpha$ be an infinite cardinal. Let $\Cal T_\alpha$ be the class of all lattices which are conditionally $\alpha$-complete and infinitely distributive. We denote by $\Cal{T}_\sigma'$ the class of all lattices $X$ such that $X$ is infinitely distributive, $\sigma$-complete and has the least element. In this paper we deal with direct factors of lattices belonging to $\Cal T_\alpha$. As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class $\Cal T_\sigma'$.
DOI : 10.21136/MB.2000.126128
Classification : 06B23, 06B35, 06D10
Keywords: direct product decomposition; infinite distributivity; conditional $\alpha$-completeness 
Jakubík, Ján. Direct product decompositions of infinitely distributive lattices. Mathematica Bohemica, Tome 125 (2000) no. 3, pp. 341-354. doi: 10.21136/MB.2000.126128
@article{10_21136_MB_2000_126128,
     author = {Jakub{\'\i}k, J\'an},
     title = {Direct product decompositions of infinitely distributive lattices},
     journal = {Mathematica Bohemica},
     pages = {341--354},
     year = {2000},
     volume = {125},
     number = {3},
     doi = {10.21136/MB.2000.126128},
     mrnumber = {1790125},
     zbl = {0967.06004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2000.126128/}
}
TY  - JOUR
AU  - Jakubík, Ján
TI  - Direct product decompositions of infinitely distributive lattices
JO  - Mathematica Bohemica
PY  - 2000
SP  - 341
EP  - 354
VL  - 125
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2000.126128/
DO  - 10.21136/MB.2000.126128
LA  - en
ID  - 10_21136_MB_2000_126128
ER  - 
%0 Journal Article
%A Jakubík, Ján
%T Direct product decompositions of infinitely distributive lattices
%J Mathematica Bohemica
%D 2000
%P 341-354
%V 125
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2000.126128/
%R 10.21136/MB.2000.126128
%G en
%F 10_21136_MB_2000_126128

[1] G. Grätzer: General Lattice Theory. Akademie Verlag, Berlin, 1972.

[2] S. S. Holland: On Radon-Nikodym Theorem in dimension lattices. Trans. Amer. Math. Soc. 108 (1963), 66-87. | DOI | MR

[3] J. Jakubík: Center of a complete lattice. Czechoslovak Math. J. 23 (1973), 125-138. | MR

[4] J. Jakubík: Center of a bounded lattice. Matem. časopis 25 (1975), 339-343. | MR

[5] J. Jakubík: Cantor-Bernstein theorem for lattice ordered groups. Czechoslovak Math. J. 22 (1972), 159-175. | MR

[6] J. Jakubík: On complete lattice ordered groups with strong units. Czechoslovak Math. J. 46 (1996), 221-230. | MR

[7] J. Jakubík: Convex isomorphisms of archimedean lattice ordered groups. Mathware Soft Comput. 5 (1998), 49-56. | MR

[8] J. Jakubík: Cantor-Bernstein theorem for complete MV-algebras. Czechoslovak Math. J. 49 (1999), 517-526. | DOI | MR

[9] J. Jakubík: Atomicity of the Boolean algebra of direct factors of a directed set. Math. Bohem. 123 (1998), 145-161. | MR

[10] J. Jakubík M. Csontóová: Convex isomorphisms of directed multilattices. Math. Bohem. 118 (1993), 359-378. | MR

[11] M. F. Janowitz: The center of a complete relatively complemented lattice is a complete sublattice. Proc. Amer. Math. Soc. 18 (1967), 189-190. | MR | Zbl

[12] J. Kaplansky: Any orthocomplemented complete modular lattice is a continuous geometry. Ann. Math. 61 (1955), 524-541. | DOI | MR | Zbl

[13] S. Maeda: On relatively semi-orthocomplemented lattices. Hiroshima Univ. J. Sci. Ser. A 24 (1960), 155-161. | MR | Zbl

[14] J. von Neumann: Continuous Geometry. Princeton Univ. Press, New York, 1960. | MR | Zbl

[15] R. Sikorski: A generalization of theorem of Banach and Cantor-Bernstein. Colloquium Math. 1 (1948), 140-144. | DOI | MR

[16] R. Sikorski: Boolean Algebras. Second Edition, Springer Verlag, Berlin, 1964. | MR | Zbl

[17] A. Tarski: Cardinal Algebras. Oxford University Press, New York, 1949. | MR | Zbl

Cité par Sources :