Article

 Title: Direct product decompositions of infinitely distributive lattices (English) Author: Jakubík, Ján Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 125 Issue: 3 Year: 2000 Pages: 341-354 Summary lang: English . Category: math . Summary: 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 $\Cal T_\sigma'$. (English) Keyword: direct product decomposition Keyword: infinite distributivity Keyword: conditional $\alpha$-completeness MSC: 06B23 MSC: 06B35 MSC: 06D10 idZBL: Zbl 0967.06004 idMR: MR1790125 . Date available: 2009-09-24T21:44:13Z Last updated: 2015-09-15 Stable URL: http://hdl.handle.net/10338.dmlcz/126128 . Reference: [1] G. Grätzer: General Lattice Theory.Akademie Verlag, Berlin, 1972. Reference: [2] S. S. Holland: On Radon-Nikodym Theorem in dimension lattices.Trans. Amer. Math. Soc. 108 (1963), 66-87. MR 0151407, 10.1090/S0002-9947-1963-0151407-3 Reference: [3] J. Jakubík: Center of a complete lattice.Czechoslovak Math. J. 23 (1973), 125-138. MR 0319831 Reference: [4] J. Jakubík: Center of a bounded lattice.Matem. časopis 25 (1975), 339-343. MR 0444537 Reference: [5] J. Jakubík: Cantor-Bernstein theorem for lattice ordered groups.Czechoslovak Math. J. 22 (1972), 159-175. MR 0297666 Reference: [6] J. Jakubík: On complete lattice ordered groups with strong units.Czechoslovak Math. J. 46 (1996), 221-230. MR 1388611 Reference: [7] J. Jakubík: Convex isomorphisms of archimedean lattice ordered groups.Mathware Soft Comput. 5 (1998), 49-56. MR 1632739 Reference: [8] J. Jakubík: Cantor-Bernstein theorem for complete MV-algebras.Czechoslovak Math. J. 49 (1999), 517-526. MR 1708370, 10.1023/A:1022467218309 Reference: [9] J. Jakubík: Atomicity of the Boolean algebra of direct factors of a directed set.Math. Bohem. 123 (1998), 145-161. MR 1673985 Reference: [10] J. Jakubík M. Csontóová: Convex isomorphisms of directed multilattices.Math. Bohem. 118 (1993), 359-378. MR 1251882 Reference: [11] M. F. Janowitz: The center of a complete relatively complemented lattice is a complete sublattice.Proc. Amer. Math. Soc. 18 (1967), 189-190. Zbl 0154.01002, MR 0200209 Reference: [12] J. Kaplansky: Any orthocomplemented complete modular lattice is a continuous geometry.Ann. Math. 61 (1955), 524-541. Zbl 0065.01801, MR 0088476, 10.2307/1969811 Reference: [13] S. Maeda: On relatively semi-orthocomplemented lattices.Hiroshima Univ. J. Sci. Ser. A 24 (1960), 155-161. Zbl 0178.33701, MR 0123494 Reference: [14] J. von Neumann: Continuous Geometry.Princeton Univ. Press, New York, 1960. Zbl 0171.28003, MR 0120174 Reference: [15] R. Sikorski: A generalization of theorem of Banach and Cantor-Bernstein.Colloquium Math. 1 (1948), 140-144. MR 0027264 Reference: [16] R. Sikorski: Boolean Algebras.Second Edition, Springer Verlag, Berlin, 1964. Zbl 0123.01303, MR 0177920 Reference: [17] A. Tarski: Cardinal Algebras.Oxford University Press, New York, 1949. Zbl 0041.34502, MR 0029954 .

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