From: Rami P Grossberg <rami+@andrew.cmu.edu>
Subject: Re: posets
Date: Thu, 16 Mar 2000 09:33:20 -0500
Newsgroups: sci.math.research
Summary: Posets with bounded chains have bounded directed subsets

>If X is a poset such that every chain has an upper bound,
>then also every directed subset has an upper bound.

>[If you think it is easy, think again.  As far as I know, it
>is easy only when X is countable.]

The uncountable case is quite easy, using induction on |X|
and the downward Lowenheim-Skolem-Tarski theorem applied to
the structure (X,<).


The original reference (not using model theory):
Tsurane Iwamura.  On Continuous Geometries I. 
Japan. J. Math. Vol 19,  57-71, 1944. 
==============================================================================

From: Rami P Grossberg <rami+@andrew.cmu.edu>
Subject: Re: posets,  correction of reference
Date: Thu, 16 Mar 2000 10:18:46 -0500
Newsgroups: sci.math.research

The result is from another paper of T. Iwamura from 1944:

A lemma on directed sets (Japanese.)
Zenkoku Shijo Sugaku Danwakai,   262,   107--111, 1944. 
==============================================================================

From: "G. A. Edgar" <edgar@math.ohio-state.edu.nospam>
Subject: Re: posets
Date: Thu, 16 Mar 2000 13:03:24 -0500
Newsgroups: sci.math.research

In article <160320000803223410%edgar@math.ohio-state.edu.nospam>, I
wrote:

> 
> If X is a poset such that every chain has an upper bound,
> then also every directed subset has an upper bound.
> 
> 

Well, I have misstated the problem.  It should be:

If X is a poset such that every chain has a least upper bound,
then also every directed subset has a least upper bound.

In article <YsoD2k600Ui702B4U0@andrew.cmu.edu>, Rami P Grossberg
<rami+@andrew.cmu.edu> wrote:

> 
> The uncountable case is quite easy, using induction on |X|
> and the downward Lowenheim-Skolem-Tarski theorem applied to
> the structure (X,<).
> 

Help me out with this.  It has been a while since I studied
logic.  I need a collection of first-order sentences, right?
"Poset"...OK I can do that.  "Every chain has an upper bound"
(or least upper bound)... naively seems to be a second-order
statement.  What next?

-- 
Gerald A. Edgar              edgar@math.ohio-state.edu