What's so special about \((\mathbb{Z}/4\mathbb{Z})^ \omega\)?
type: Item
instance of: scholarly article
title: What's so special about \((\mathbb{Z}/4\mathbb{Z})^ \omega\)?
zbMATH Open document ID: 0782.03016
DOI: 10.1007/BF01387764
author: Gisela Ahlbrandt
author: Martin Ziegler
published in: Archive for Mathematical Logic
publication date: 1993-08-25T00:00:00Z
review text: It is known that any totally categorical structure is built up from a strictly minimal set as a sequence of covers. As the strictly minimal sets in totally categorical structures are classified, to classify totally categorical structures one needs to understand covers. A structure \(M\) is said to be a cover of its 0-definable subset \(N\) if (1) every definable relation on \(N\) is definable with parameters in \(N\) and (2) there are a 0-definable surjection \(f: M\backslash N\to N\) and a definable family \(\{G_ a: a\in N\}\) of groups living in \(N^{eq}\) which act regularly on the fibers \(f^{-1}(a)\) in a definable way. For any structure \(N\) with a definable family \(\{G_ a: a\in N\}\) of groups living in \(N^{eq}\), every cover is a certain expansion of the so-called principal cover \(M_ 0\) which is obtained from \(N\) by adding for every \(a\in A\) a new set \(F_ a\) with a regular action of \(G_ a\) on it and has no other structure. The cover problem is to determine all covering expansions of \(M_ 0\) which induce a new structure on \(N\). The problem turned out to be difficult even in special simple cases. The authors consider the case when \(N\) is the set of nonzero elements of \(\mathcal V\), a countably dimensional vector space over \(\mathbb{F}_ 2\). They solve the cover problem in two special cases: all the groups \(G_ a\) are \((\mathbb{F}_ 2,+)\) (finite type) and each \(G_ a\) is \({\mathcal V}/\mathbb{F}_{2a}\) (affine type). It turns out that all possible covers in the finite case are in some sense `trivial'; in the affine case there are only two `nontrivial' covers which come from the group \((\mathbb{Z}/4\mathbb{Z})^ \omega\), a favorite example in model theory; therefore the title. For countable totally categorical \(N\), the cover problem can be restated in algebraic terms as follows: determine all closed subgroups \(\mathcal G\) of \(\Aut(M_ 0)\) for which the projection map \({\mathcal G}\to\Aut(M_ 0)\) is surjective. In the special case…
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 179620
zbMATH Keywords: totally categorical structure
zbMATH Keywords: covers
reviewed by: Oleg Belegradek
MaRDI profile type: Publication
cites work: Quasi finitely axiomatizable totally categorical theories
cites work: Q3994342
cites work: On the cohomology of the finite special linear groups. I, II
cites work: \(\aleph _ 0\)-categorical, \(\aleph _ 0\)-stable structures
cites work: Q5648522
cites work: Cohomology of Structures and Some Problems of Ahlbrandt and Ziegler
cites work: Totally Categorical Structures
cites work: Q4735180
cites work: The uniqueness of envelopes in ℵ0-categorical, ℵ0-stable structures
description: scientific article; zbMATH DE number 179620