Chains and directed sets
Date: Wed, 27 Nov 91 15:06:49 -0500
To: firstname.lastname@example.org (Val Breazu-Tannen)
In-reply-to: your message of Tue, 26 Nov 91 19:29:47 EST
For domain theory, the notion of isolated elements and
algebracity is important. One has to be careful about
the cardinality problem here.
For example, the power set of the real ordered by inclusion
is algebraic in terms of directed sets, but NOT in terms
of chains. The isolated elements are the finite subsets
(in both cases), but a chain of finite elements is just
not enough to approximate an uncountable set.
So it seems to me when Iwamura showed chain-complete
implies directed-complete, algebracity was not a concern.
It seems Iwamura's result is less relevant here.