The structure of saturated free algebras
Let M be a free algebra with a countable similarity type and suppose that M is \(\aleph_ 1-saturated\), then M has a finite dimensional \(\omega\)- stable theory, precisely: Theorem 1. There is a finite set of regular types \(q_ 1,q_ 2,...,q_ k\) such that every model N of Th(M) is prime over a union \(Y_ 1\cup...\cup Y_ k\) of indiscernible sets \(Y_ i\), each based on the corresponding \(q_ i\). (With the consequence that Th(M) has at most \(\alpha +\aleph_ o\) models of cardinality \(\aleph_{\alpha}\geq 2^{\aleph_ o}.)\) The authors announce also the following more algebraic result (which relies on Theorem 1), but their proof is quite incomplete: Theorem 2. There is a finite set of types of weight one, say \(q_ 1,...,q_ k,\) such that each model N of Th(M) is generated, in the usual algebraic sense, by a union \(Y_ 1\cup...\cup Y_ k\) of indiscernible sets \(Y_ i\), each based on the corresponding \(q_ i.\) Concrete examples are given, illustrating for example the difference between the notions of regular types and of types of weight one.
scientific article; zbMATH DE number 3853047
scientific article