The structure of saturated free algebras
type: Item
instance of: scholarly article
title: The structure of saturated free algebras
zbMATH Open document ID: 0537.03020
DOI: 10.1007/BF01194528
author: John T. Baldwin
author: S. Shelah
published in: Algebra Universalis
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C05
zbMATH DE Number: 3853047
zbMATH Keywords: saturated algebras
zbMATH Keywords: weight one types
zbMATH Keywords: free algebra
zbMATH Keywords: regular types
zbMATH Keywords: indiscernible sets
reviewed by: Chantal Berline
MaRDI profile type: Publication
cites work: A combinatorial fact about free algebras
cites work: Counting models in universal Horn classes
cites work: Classification theory and the number of non-isomorphic models
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description: scientific article; zbMATH DE number 3853047