Constructions of many complicated uncountable structures and Boolean algebras

Abstract

This article has three aims: (1) To make the results of [12, VIII] on constructing models more available for application, by separating the combinatorial parts. Thus in applications one will only need the relevant things from the area of application. (2) To strengthen the results there. In particular, we were mainly interested in [12, VIII] in showing that there are many isomorphism types of models of an unsuperstable theory, with results about the number of models not elementarily embeddable in each other being a side benefit. Here we consider the latter case in more detail, getting more cases. We also consider some more complicated constructions along the same lines % MathType!MTEF!2!1!+-$(K_{ptr}^\omega )$. (3) To solve various problems from the list of van Dowen, Monk and Rubin [3] on Boolean algebras, which was presented at a conference on Boolean algebra in Oberwolfach January 1979 (most of the solutions are mentioned in the final version). Some of them are not related to (1) and (2). This continues [10, §2] in which the existence of a rigid B.A. in every uncountable power was proved. There (and also here) we want to demonstrate the usefulness of the methods developed in [12, VIII] (and §§ 1,2) for getting many (rigid) non-embeddable models in specific classes.