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Comment to Zhang re Chains and directed sets; reply by Zhang



Date: Sun, 1 Dec 91 11:11:36 EST
To: gqz@engin.umich.edu (Guo-Qiang Zhang)
Cc: gdp@dcs.edinburgh.ac.uk

Rereading your message to the types mailing list, I wonder if you
don't write "chain" when you mean "omega-chain".  If "chain" can have
any cardinality then your remark:

"Plotkin in his `Pisa  Notes'  indicated that for  omega-algebraic cpos
chains and directed sets are interchangeable."

[NOTE: in the Pisa notes, the definition of cpo requires only
omega-chains to have lubs]

is misleading since as far as I can tell now, there is no statement in
the Pisa notes that says "in an omega-algebraic cpo if arbitrary
chains have lubs then arbitrary directed subsets have lubs " (or
something analoguous for function continuity). In view of Iwamura's
result, omega-algebraicity would be superfluous in such a statement.
Unfortunately, I took your message to mean just that and I posted mine
without trying to find the statement in the Pisa notes. My apologies
to Gordon Plotkin!

If however, your "chain" means "omega-chain" then you may be referring
to the result in the Pisa notes that in an omega-algebraic cpo
arbitrary directed subsets have lubs. This, of course, is an
interesting fact that is separate from Iwamura's.

Val Tannen

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Date: Sun, 1 Dec 91 13:47:58 -0500
From: Guoqiang Zhang <gqz@engin.umich.edu>
To: types@theory.lcs.mit.edu

To Val Tannen
-------------

>If however, your "chain" means "omega-chain" then you may be referring
>to the result in the Pisa notes that in an omega-algebraic cpo
>arbitrary directed subsets have lubs. This, of course, is an
>interesting fact that is separate from Iwamura's.

----- The same would be  true for continuous  functions.  

----- So   you claimed my  statements   in  the posting  `superfluous'
without even wondering if omega-chains were used.   In fact, when John
Mitchell posted    the original   question   which  prompted  all  the
discussion, he  was clearly  referring  to domain  theory,  where only
omega-chains are used most of the time.  It is only after your message
which reminded people of the Iwamura-Markowski result that the general
notion of chains  was   brought into context (which  caused  a  lot of
confusion).

I was   wondering  if linearly ordered   sets   were used in Plotkin's
statement  instead  of   just  omega-chains, why would it  be NOT
interesting?

I do agree that in the Pisa notes Plotkin uses omega-chains.  However,
in my example of the powerset of the real  posted in the types mailing
list, if you take chains to be  linearly ordered sets,  the example is
still valid.  Note that finite elements will  still be finite subsets,
and you simply cannot form  a chain larger  than an  omega-chain using
only the finite subsets. Therefore, you still cannot approximate an
uncountable set by a chain of finite subsets.


Guo-Qiang Zhang