Date: Wed, 18 Oct 1995 10:36:21 -0300 (ADT)
Subject: a question

Date: Wed, 18 Oct 1995 09:33:34 +0100
From: TONOLO@PDMAT1.MATH.UNIPD.IT

Dear Professors,

I wish to know if complete lattices with the following property have
been considered and named:

         h < sup a_i  ==>   h = sup( h : a_i)      (I use : for the binary meet)
for each h and each directed family {a_i | i\in I} of elements of the lattice.


Thanking for the attention, with my best regards

Alberto Tonolo


Date: Wed, 18 Oct 1995 11:43:43 -0300 (ADT)
Subject: Re: a question

Date: Wed, 18 Oct 1995 14:22:17 +0000
From: Steve Vickers <sjv@doc.ic.ac.uk>

>I wish to know if complete lattices with the following property have
>been considered and named:
>
>         h < sup a_i  ==>   h = sup( h : a_i)      (I use : for the binary
>meet)
>for each h and each directed family {a_i | i\in I} of elements of the lattice.
The condition is equivalent to binary meets distributing over directed
joins, and it's well known (e.g. Johnstone's "Stone Spaces" Lemma VII.4.1)
that this holds for all continuous lattices.

Posets that have finite meets and directed joins and this distributivity
have been called "preframes" by Banaschewski, so if you are genuinely
interested in complete lattices with preframe distributivity you might
plausibly call them complete preframes.

Steve Vickers.


Date: Wed, 18 Oct 1995 16:03:07 -0300 (ADT)
Subject: Re: a question

Date: Wed, 18 Oct 1995 14:34:11 -0400 (EDT)
From: MTHISBEL@ubvms.cc.buffalo.edu

<<I wish to know if complete lattices with the following property have
been considered and named:
    h < sup a_i  ==>   h = sup( h : a_i)      (I use : for the binary meet)
for each h and each directed family {a_i | i\in I} of elements of the lattice.>>

  You have the meet-continuous lattices introduced in 1948 by G. Birkhoff
and O. Frink. The usual definition is, for each up-directed family {a_i}
and each h,h : sup a_i = sup h : a_i (in your notation). This is easily
seen to be equivalent. There is a lot known about them by now.
         Yours,    John Isbell



Date: Fri, 20 Oct 1995 10:37:15 -0300 (ADT)
Subject: Re: "a question"

Date: 19-OCT-1995 19:34:07.42
From: Fred E.J. Linton <FLINTON@EAGLE.WESLEYAN.EDU>

Alberto Tonolo asks about complete lattices with the following property
(I replace his meet-symbol ":" with the symbol "^" instead):

>            h < sup a_i  ==>
>
>            h = sup( h ^ a_i)                 (*_h)
>
> for each h and each directed family {a_i | i in I} of elements of the lattice.

I can point out only that, in any lattice (complete
or not) in which the family  a_i  (i in I)  has a sup,
to require  (*_h)  for each  h < sup a_i  is the same as
to require the distributivity

             h ^ sup a_i = sup (h ^ a_i)       (d_h)

for just all  h .

Indeed, the relations  (d_h)  and  (*_h)  coincide, and are trivial,
when  h < sup a_i ; and when  h = sup a_i  they not only coincide but
are tautologous.  In particular,  (d_h)  for all  h  guarantees  (*_h)
for  h < sup a_i ,  and  (*_h)  for  h < sup a_i  guarantees  (d_h)
at least for all  h  that are  <  or  =  to  sup a_i .

For general  h' , let  h = h' ^ sup a_i .  Then  either  h < sup a_i
or  h = sup a_i ; in either case,  (d_h)  holds.  To see that  (d_h')
holds as well, just calculate:

        since h = h' ^ sup a_i ,  either  h <  or  h = sup a_i  ;

and, by  (*_h) ,

        h' ^ sup a_i = h = sup (h ^ a_i) = sup ((h' ^ sup a_i) ^ a_i) =

                     = sup (h' ^ ((sup a_i) ^ a_i)) = sup (h' ^ a_i) .

In particular, *every*  h ^ (-)  will distribute through whatever joins
happen to satisfy  (*_h)  for  just all smaller  h  than the join;
if the lattice is complete and satisfies the (*_h) condition for all
joins and all  h  smaller than them, it is a frame; and, I guess, if
I'm not utterly confused by what the term "continuous lattice" means,
if the lattice has directed joins and the (*_h) condition holds for all
directed joins and all  h  smaller than them, then it's a continuous lattice
(or, anyway, *every*  h ^ (-)  distributes through every directed join).

Hope this helps.  And I'm sure someone will correct me if I've misunderstood
the meaning of "continuous lattice" :-) .

-- FEJ

[PS: Address change: please make a note of it --

  the address  fejlinton @ attmail.com  is officially dead

(AT&T Mail's recent trebling of their monthly fee made me leave their service).

Still valid are  FLinton @ Wesleyan.EDU , the older and more complicated
 FLinton @ eagle.Wesleyan.EDU , the still older  FLinton @ WESLEYAN.bitnet ,
and  fejlinton @ mcimail.com .

Thanks for your attention.     - FEJ ]


Date: Mon, 23 Oct 1995 11:34:26 -0300 (ADT)
Subject: Re: "a question"

Date: Mon, 23 Oct 1995 09:36:12 EDT5EST
From: Philipp Sunderhauf <PSUNDER@payson.usm.maine.edu>

Fred E.J. Linton in an answer to the question of Alberto Tonolo:

> I guess, if
> I'm not utterly confused by what the term "continuous lattice" means,
> if the lattice has directed joins and the (*_h) condition holds for all
> directed joins and all  h  smaller than them, then it's a continuous lattice
> (or, anyway, *every*  h ^ (-)  distributes through every directed
> join).
> Hope this helps.  And I'm sure someone will correct me if I've misunderstood
> the meaning of "continuous lattice" :-) .

1) "continuous lattice" refers to complete lattices only, so the
condition of existing directed sups is obsolete.

2) The equational characterisation of continuous lattices is directed
sups distributing over *arbitrary* infs. As pointed out by John
Isbell, the lattices referred to by Alberto Tonolo are known as "meet-
continuous".

3) Detailed answers to this and related questions may be found in
The Compendium:
G Gierz, KH Hofmann, K Keimel, JD Lawson, M Mislove, D Scott:
A Compendium of Continuous Lattices. Springer 1980.

Philipp.
------
Philipp S"underhauf
Department of Mathematics and Statistics
University of Southern Maine
Portland, ME 04103
psunder@usm.maine.edu


Date: Mon, 23 Oct 1995 11:33:27 -0300 (ADT)
Subject: Re: a question

Date: Sun, 22 Oct 1995 11:20:07 -0700
From: William H. Rowan <rowan@crl.com>

Dear Alberto,

It would seem your property is implied by one form of continuity, I think
it is upper continuity.  I think algebraic lattices also have your property.

Best,

Bill Rowan