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.

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