# On the injectivity of Boolean algebras

Commentationes Mathematicae Universitatis Carolinae (1993)

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

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topBanaschewski, 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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