Direct product decompositions of infinitely distributive lattices

Ján Jakubík

Mathematica Bohemica (2000)

  • Volume: 125, Issue: 3, page 341-354
  • ISSN: 0862-7959

Abstract

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Let α be an infinite cardinal. Let 𝒯 α be the class of all lattices which are conditionally α -complete and infinitely distributive. We denote by 𝒯 σ ' the class of all lattices X such that X is infinitely distributive, σ -complete and has the least element. In this paper we deal with direct factors of lattices belonging to 𝒯 α . As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class 𝒯 σ ' .

How to cite

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Jakubík, Ján. "Direct product decompositions of infinitely distributive lattices." Mathematica Bohemica 125.3 (2000): 341-354. <http://eudml.org/doc/248679>.

@article{Jakubík2000,
abstract = {Let $\alpha $ be an infinite cardinal. Let $\mathcal \{T\}_\alpha $ be the class of all lattices which are conditionally $\alpha $-complete and infinitely distributive. We denote by $\mathcal \{T\}_\sigma ^\{\prime \}$ 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 $\mathcal \{T\}_\alpha $. As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class $\mathcal \{T\}_\sigma ^\{\prime \}$.},
author = {Jakubík, Ján},
journal = {Mathematica Bohemica},
keywords = {direct product decomposition; infinite distributivity; conditional $\alpha $-completeness; direct product decomposition; infinite distributivity; conditional -completeness},
language = {eng},
number = {3},
pages = {341-354},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Direct product decompositions of infinitely distributive lattices},
url = {http://eudml.org/doc/248679},
volume = {125},
year = {2000},
}

TY - JOUR
AU - Jakubík, Ján
TI - Direct product decompositions of infinitely distributive lattices
JO - Mathematica Bohemica
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 125
IS - 3
SP - 341
EP - 354
AB - Let $\alpha $ be an infinite cardinal. Let $\mathcal {T}_\alpha $ be the class of all lattices which are conditionally $\alpha $-complete and infinitely distributive. We denote by $\mathcal {T}_\sigma ^{\prime }$ 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 $\mathcal {T}_\alpha $. As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class $\mathcal {T}_\sigma ^{\prime }$.
LA - eng
KW - direct product decomposition; infinite distributivity; conditional $\alpha $-completeness; direct product decomposition; infinite distributivity; conditional -completeness
UR - http://eudml.org/doc/248679
ER -

References

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  1. G. Grätzer, General Lattice Theory, Akademie Verlag, Berlin, 1972. (1972) 
  2. S. S. Holland, 10.1090/S0002-9947-1963-0151407-3, Trans. Amer. Math. Soc. 108 (1963), 66-87. (1963) MR0151407DOI10.1090/S0002-9947-1963-0151407-3
  3. J. Jakubík, Center of a complete lattice, Czechoslovak Math. J. 23 (1973), 125-138. (1973) MR0319831
  4. J. Jakubík, Center of a bounded lattice, Matem. časopis 25 (1975), 339-343. (1975) MR0444537
  5. J. Jakubík, Cantor-Bernstein theorem for lattice ordered groups, Czechoslovak Math. J. 22 (1972), 159-175. (1972) MR0297666
  6. J. Jakubík, On complete lattice ordered groups with strong units, Czechoslovak Math. J. 46 (1996), 221-230. (1996) MR1388611
  7. J. Jakubík, Convex isomorphisms of archimedean lattice ordered groups, Mathware Soft Comput. 5 (1998), 49-56. (1998) MR1632739
  8. J. Jakubík, 10.1023/A:1022467218309, Czechoslovak Math. J. 49 (1999), 517-526. (1999) MR1708370DOI10.1023/A:1022467218309
  9. J. Jakubík, Atomicity of the Boolean algebra of direct factors of a directed set, Math. Bohem. 123 (1998), 145-161. (1998) MR1673985
  10. J. Jakubík M. Csontóová, Convex isomorphisms of directed multilattices, Math. Bohem. 118 (1993), 359-378. (1993) MR1251882
  11. M. F. Janowitz, The center of a complete relatively complemented lattice is a complete sublattice, Proc. Amer. Math. Soc. 18 (1967), 189-190. (1967) Zbl0154.01002MR0200209
  12. J. Kaplansky, 10.2307/1969811, Ann. Math. 61 (1955), 524-541. (1955) Zbl0065.01801MR0088476DOI10.2307/1969811
  13. S. Maeda, On relatively semi-orthocomplemented lattices, Hiroshima Univ. J. Sci. Ser. A 24 (1960), 155-161. (1960) Zbl0178.33701MR0123494
  14. J. von Neumann, Continuous Geometry, Princeton Univ. Press, New York, 1960. (1960) Zbl0171.28003MR0120174
  15. R. Sikorski, 10.4064/cm-1-2-140-144, Colloquium Math. 1 (1948), 140-144. (1948) MR0027264DOI10.4064/cm-1-2-140-144
  16. R. Sikorski, Boolean Algebras, Second Edition, Springer Verlag, Berlin, 1964. (1964) Zbl0123.01303MR0177920
  17. A. Tarski, Cardinal Algebras, Oxford University Press, New York, 1949. (1949) Zbl0041.34502MR0029954

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