Inverse of Heyting algebra morphism is p-morphism
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It is known that the category of Heyting algebras is dually equivalent to the category of Esakia spaces (which is equivalent to the category of descriptive intuitionistic Kripke frames). Under the duality, a Heyting morphism $f : H to H'$ is send to its inverse, that is it sends a prime filter in $H'$ to its inverse image.
Question: Why is this inverse map a p-morphism (or bounded morphism)?
All references I have found for this refer to a paper by Esakia himself, in Russian. I am looking for an English reference or a proof.
reference-request heyting-algebra
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
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add a comment |
$begingroup$
It is known that the category of Heyting algebras is dually equivalent to the category of Esakia spaces (which is equivalent to the category of descriptive intuitionistic Kripke frames). Under the duality, a Heyting morphism $f : H to H'$ is send to its inverse, that is it sends a prime filter in $H'$ to its inverse image.
Question: Why is this inverse map a p-morphism (or bounded morphism)?
All references I have found for this refer to a paper by Esakia himself, in Russian. I am looking for an English reference or a proof.
reference-request heyting-algebra
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
$endgroup$
1
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Check out Lemma 5.5 of Patrick Morandi's excellent notes on duality theory for distributive lattices and Heyting algebras.
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– FML
Jan 7 at 12:36
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I'd found them indeed. Thanks!
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– Math Student 020
Jan 8 at 0:48
1
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For a somewhat different type of argument you can also have a look at Theorem 6 of the paper Canonical extensions, Esakia spaces, and universal models by Mai Gehrke.
$endgroup$
– FML
Jan 8 at 9:34
add a comment |
$begingroup$
It is known that the category of Heyting algebras is dually equivalent to the category of Esakia spaces (which is equivalent to the category of descriptive intuitionistic Kripke frames). Under the duality, a Heyting morphism $f : H to H'$ is send to its inverse, that is it sends a prime filter in $H'$ to its inverse image.
Question: Why is this inverse map a p-morphism (or bounded morphism)?
All references I have found for this refer to a paper by Esakia himself, in Russian. I am looking for an English reference or a proof.
reference-request heyting-algebra
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
$endgroup$
It is known that the category of Heyting algebras is dually equivalent to the category of Esakia spaces (which is equivalent to the category of descriptive intuitionistic Kripke frames). Under the duality, a Heyting morphism $f : H to H'$ is send to its inverse, that is it sends a prime filter in $H'$ to its inverse image.
Question: Why is this inverse map a p-morphism (or bounded morphism)?
All references I have found for this refer to a paper by Esakia himself, in Russian. I am looking for an English reference or a proof.
reference-request heyting-algebra
reference-request heyting-algebra
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
asked Dec 17 '18 at 5:42
Math Student 020Math Student 020
956616
956616
1
$begingroup$
Check out Lemma 5.5 of Patrick Morandi's excellent notes on duality theory for distributive lattices and Heyting algebras.
$endgroup$
– FML
Jan 7 at 12:36
$begingroup$
I'd found them indeed. Thanks!
$endgroup$
– Math Student 020
Jan 8 at 0:48
1
$begingroup$
For a somewhat different type of argument you can also have a look at Theorem 6 of the paper Canonical extensions, Esakia spaces, and universal models by Mai Gehrke.
$endgroup$
– FML
Jan 8 at 9:34
add a comment |
1
$begingroup$
Check out Lemma 5.5 of Patrick Morandi's excellent notes on duality theory for distributive lattices and Heyting algebras.
$endgroup$
– FML
Jan 7 at 12:36
$begingroup$
I'd found them indeed. Thanks!
$endgroup$
– Math Student 020
Jan 8 at 0:48
1
$begingroup$
For a somewhat different type of argument you can also have a look at Theorem 6 of the paper Canonical extensions, Esakia spaces, and universal models by Mai Gehrke.
$endgroup$
– FML
Jan 8 at 9:34
1
1
$begingroup$
Check out Lemma 5.5 of Patrick Morandi's excellent notes on duality theory for distributive lattices and Heyting algebras.
$endgroup$
– FML
Jan 7 at 12:36
$begingroup$
Check out Lemma 5.5 of Patrick Morandi's excellent notes on duality theory for distributive lattices and Heyting algebras.
$endgroup$
– FML
Jan 7 at 12:36
$begingroup$
I'd found them indeed. Thanks!
$endgroup$
– Math Student 020
Jan 8 at 0:48
$begingroup$
I'd found them indeed. Thanks!
$endgroup$
– Math Student 020
Jan 8 at 0:48
1
1
$begingroup$
For a somewhat different type of argument you can also have a look at Theorem 6 of the paper Canonical extensions, Esakia spaces, and universal models by Mai Gehrke.
$endgroup$
– FML
Jan 8 at 9:34
$begingroup$
For a somewhat different type of argument you can also have a look at Theorem 6 of the paper Canonical extensions, Esakia spaces, and universal models by Mai Gehrke.
$endgroup$
– FML
Jan 8 at 9:34
add a comment |
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1
$begingroup$
Check out Lemma 5.5 of Patrick Morandi's excellent notes on duality theory for distributive lattices and Heyting algebras.
$endgroup$
– FML
Jan 7 at 12:36
$begingroup$
I'd found them indeed. Thanks!
$endgroup$
– Math Student 020
Jan 8 at 0:48
1
$begingroup$
For a somewhat different type of argument you can also have a look at Theorem 6 of the paper Canonical extensions, Esakia spaces, and universal models by Mai Gehrke.
$endgroup$
– FML
Jan 8 at 9:34