On the injectivity of Boolean algebras

Bernhard Banaschewski

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 3, page 501-511
  • ISSN: 0010-2628

Abstract

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The functor taking global elements of Boolean algebras in the topos 𝐒𝐡 𝔅 of sheaves on a complete Boolean algebra 𝔅 is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in 𝔅 -valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.

How to cite

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Banaschewski, Bernhard. "On the injectivity of Boolean algebras." Commentationes Mathematicae Universitatis Carolinae 34.3 (1993): 501-511. <http://eudml.org/doc/247519>.

@article{Banaschewski1993,
abstract = {The functor taking global elements of Boolean algebras in the topos $\text\{$\mathbf \{Sh\}\mathfrak \{B\}$\}$ of sheaves on a complete Boolean algebra $\mathfrak \{B\}$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\mathfrak \{B\}$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.},
author = {Banaschewski, Bernhard},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {sheaves on a complete Boolean algebra; injective Boolean algebra; complete Boolean algebra; injective complete Boolean algebra; absolute frame retract; Boolean-valued set theory; topos of sheaves on a complete Boolean algebra; frame retract; global elements; injectivity; completeness; Boolean Ultrafilter Theorem; category of complete Boolean algebras; injectives; absolute retracts},
language = {eng},
number = {3},
pages = {501-511},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the injectivity of Boolean algebras},
url = {http://eudml.org/doc/247519},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Banaschewski, Bernhard
TI - On the injectivity of Boolean algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 3
SP - 501
EP - 511
AB - The functor taking global elements of Boolean algebras in the topos $\text{$\mathbf {Sh}\mathfrak {B}$}$ of sheaves on a complete Boolean algebra $\mathfrak {B}$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\mathfrak {B}$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.
LA - eng
KW - sheaves on a complete Boolean algebra; injective Boolean algebra; complete Boolean algebra; injective complete Boolean algebra; absolute frame retract; Boolean-valued set theory; topos of sheaves on a complete Boolean algebra; frame retract; global elements; injectivity; completeness; Boolean Ultrafilter Theorem; category of complete Boolean algebras; injectives; absolute retracts
UR - http://eudml.org/doc/247519
ER -

References

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  1. Banaschewski B., On pushing out frames, Comment. Math. Univ. Carolinae 31 (1990), 13-21. (1990) Zbl0706.18003MR1056165
  2. Banaschewski B., Bhutani K.R., Boolean algebras in a localic topos, Math. Proc. Cambridge Phil. Soc. 100 (1986), 43-55. (1986) Zbl0598.18001MR0838652
  3. Bell J.L., On the strength of the Sikorski Extension Theorem for Boolean algebras, J. Symb. Logic 48 (1983), 841-846. (1983) Zbl0537.03032MR0716646
  4. Blass A., Sčedrov A., Freyd's models for the independence of the axiom of choice, Mem. Amer. Math. Soc. 79 (1989), No. 404. (1989) Zbl0687.03031
  5. Higgs D., A category approach to Boolean-valued set theory, preprint, University of Waterloo, 1973. 
  6. Johnstone P.T., Topos theory, L.M.S. Mathematical Monographs no. 10, Academic Press, 1977. Zbl1071.18002MR0470019
  7. Johnstone P.T., Conditions related to De Morgan's law, Springer LNM 253 (1979), 479-491. Zbl0445.03041MR0555556
  8. Johnstone P.T., Stone spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. Zbl0586.54001MR0698074
  9. Mac Lane S., Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1971. Zbl0906.18001MR0354798
  10. Pultr A., Oral communication, October 1986, . 

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