### More on entangled orders

by Shafir and Shelah. [SaSh:553]

J Symbolic Logic, 2000

This paper grew as a continuation of [Sh462] but in the present
form it can serve as a motivation for it as well. We deal
with
the
same notions, and use just one simple lemma from there. Originally
entangledness was introduced in order to get narrow Boolean
algebras
and examples of the nonmultiplicativity of c.c-ness. These
applications became marginal when the hope to extract new
such
objects or strong colourings were not materialized, but after
the
pcf constructions which made their debut in [Sh:g] it seems
that
this notion gained independence. Generally we aim at characterizing
the existence strong and weak entangled orders in cardinal
arithmetic terms. In [Sh462] necessary conditions were shown
for
strong entangledness which in a previous version was erroneously
proved to be equivalent to plain entangledness. In section
1
we give
a forcing counterexample to this equivalence and in section
2
we get
those results for entangledness (certainly the most interesting
case). In section 3 we get weaker results for positively
entangledness,
especially when supplemented with the existence of a separating
point. An antipodal case is defined and completely characterized.
Lastly we outline a forcing example showing that these two
subcases
of positive entangledness comprise no dichotomy.

Back to the list of publications