- Dear Tadeusz,

Do you have references for Esakia's proof of the result you mentioned

(i.e. that if in a complete Heyting algebra every element is a join of

completely join irreducibles then the cHa is spatial)?

Results like this came to me as part of the computer science folklore of

algebraic dcpos (directed complete partial orders), and their origin is

obscure to me.

Mind you, with hindsight I suspect it's not hard to deduce this

particular result from Raney's work (1953) on completely distributive

lattices. (Completely distributive lattices are topologies for the more

general continuous dcpos.)

On a related point, your characterization of spatiality (every element

is an inf of meet irreducibles) for complete Heyting algebras is not

constructively valid. Constructively you need to find points as

completely prime filters instead of meet irreducibles. (Classically they

are in bijection - given a completely prime filter, take the join of the

elements not in it.) Then the criterion becomes that, for any elements a

and b, if every completely prime filter containing a also contains b

then a <= b.

Regards,

Steve Vickers. - Dear Professor Vickers,

I'm very obliged for your e-mail, for prof. Kearnes' lucid answer (btw, is this

theorem a part of the folklore or can it be attributed to a particular author?)

to my main problem and for prof. Resende's terminological comment. Sorry I

couldn't answer ealier, I was absent.

Concerning your question about references for the proof of the theorem I

attributed to prof. Esakia: I found it in his book 'Algebry Heytinga I. Teoria

dvoistvennosti' ('Heyting algebras I. Duality Theory') published in Tbilisi,

1985 (in Russian). I am not sure if Leo Esakia himself would claim the theorem

his own. The references to the suitable chapter mention 'interesting and deep

results related to superintuitionistic logics [logical counterpart of the theory

of Heyting algebras, roughly speaking] obtained by A. Kuznetsov and his

students'; the only paper mentioned by name is Kuznetsov's work from 1971. Some

of the results from Esakia's book can be also found in his earlier paper,

published in 1974 in Soviet Mathematics Doklady.

Actually, I wouldn't be that much surprised if some of the results known now as

'part of the computer science folklore of algebraic dcpos' (btw, could you

recommend me a good handbook or monograph of the subject? this Raney's work or

something newer?) turned out to have originated in Kuznetsov's school. I have,

however, no material to support such claims, so my guess can be wrong. If you

want to investigate that, you can contact Leo Esakia by mail:

esakia@.... It is also not unlikely that such theorems could have been

proven independently in various centres. Communication between the two sides of

the Iron Curtain was far from being smooth.

Your comment on a constructive way of defining spatiality raises an interesting

point but the thing is that I'm rather interested in the theory of (varieties

of) Heyting algebra and metatheory of extensions of intuitionistic logics from a

classical point of view. It seems to me that in this field you cannot insist too

much on being constructive. Even most basic theorems like Birkhoff's 1944

theorem that every algebra is a subdirect product of subdirect irreducibilities

or Jonsson's 1967 lemma on congruence-distributive varieties use some form of

Zorn's lemma or the Axiom of Choice.

Actually, the main reason why I posted my question was my interest in an old

problem posed by Kuznetsov: for every variety of Heyting algebras, can every

equation which fails to hold be refuted in a spatial locale from the variety? If

there are any theorems e.g., of the computer science folklore of algebraic dcpos

which can be helpful in answering this question, I would be very obliged for any

information.

Best regards,

Tadeusz Litak

>X-UIDL: ^Mk!!3ZL"!cAV!!0p9!!

Gecko/20030312

>Date: Mon, 01 Dec 2003 10:16:25 +0000

>From: Steve Vickers <s.j.vickers@...>

>User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.0; en-US; rv:1.3)

>X-Accept-Language: en-us, en

>MIME-Version: 1.0

>To: LITAK Tadeusz Michal <litak@...>

>CC: univalg@yahoogroups.com

>Subject: Leo Esakia

>Content-Transfer-Encoding: 7bit

>

>Dear Tadeusz,

>

>Do you have references for Esakia's proof of the result you mentioned

>(i.e. that if in a complete Heyting algebra every element is a join of

>completely join irreducibles then the cHa is spatial)?

>

>Results like this came to me as part of the computer science folklore of

>algebraic dcpos (directed complete partial orders), and their origin is

>obscure to me.

>

>Mind you, with hindsight I suspect it's not hard to deduce this

>particular result from Raney's work (1953) on completely distributive

>lattices. (Completely distributive lattices are topologies for the more

>general continuous dcpos.)

>

>On a related point, your characterization of spatiality (every element

>is an inf of meet irreducibles) for complete Heyting algebras is not

>constructively valid. Constructively you need to find points as

>completely prime filters instead of meet irreducibles. (Classically they

>are in bijection - given a completely prime filter, take the join of the

>elements not in it.) Then the criterion becomes that, for any elements a

>and b, if every completely prime filter containing a also contains b

>then a <= b.

>

>Regards,

>

>Steve Vickers.

>