Powered by OpenAIRE graph
Found an issue? Give us feedback
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Journal of Symbolic ...arrow_drop_down
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
Journal of Symbolic Logic
Article . 1981 . Peer-reviewed
License: Cambridge Core User Agreement
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 1981
Data sources: zbMATH Open
DBLP
Article . 2017
Data sources: DBLP
versions View all 3 versions
addClaim

Preservation of saturation and stability in a variety of nilpotent groups

Authors: Pat Rogers;

Preservation of saturation and stability in a variety of nilpotent groups

Abstract

This paper is a contribution to the growing literature on the model theory of nilpotent groups. (See Baumslag and Levin [2]; Eršov [5]; Hodges [9], [10]; Mal′cev [14]; Olin [16] and Saracino [19], [20].) In it we investigate the conditions under which the free product in the variety of all nilpotent of class 2 (nil-2) groups preserves saturation and stability.It is well known that the direct product preserves both saturation (see Waszkiewicz and Wȩglorz [23]) and stability (see Wierzejewski [24]; Macintyre [13]; Eklof and Fisher [4]). On the other hand it is easy to show that the full free product of groups preserves neither property; indeed, in the case of saturation this failure is extremely bad since no free product of nontrivial groups is even 2-saturated. Our results show that the nil-2 free product falls between these two extremes.Our proofs are mainly model-theoretic with a smattering of elementary algebra and rely heavily upon the unique normal form for the elements of a nil-2 free product given by MacHenry in [12]. (This normal form and some of its consequences are discussed in §1.) We assume familiarity with the basic ideas of saturation (see Chapter 5 of [3]) and Shelah's treatment of stability in [22].We prove two main theorems in §3 each giving a necessary and sufficient condition in separate situations for the preservation of saturation. In the first (Theorem 3.1) we allow one finite factor, while in the second (Theorem 3.10) we deal solely with torsion groups. Our motivation for the proof of sufficiency was the paper of Waszkiewicz and Wȩglorz [23] and the principal tool is a “Feferman-Vaught” Theorem for the nil-2 free product which we prove in §2. We also show that if both factors in a nil-2 free product are nontorsion and one factor has a nil-2 basis, then the group is not even 3-saturated. We leave open the case where both factors are infinite but only one is torsion.

Related Organizations
Keywords

Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, nil-2 free product, instability results, Model-theoretic algebra, Feferman-Vaught pair of groups, Classification theory, stability, and related concepts in model theory, nil-2 group

  • BIP!
    Impact byBIP!
    selected citations
    These citations are derived from selected sources.
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    3
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Top 10%
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
Powered by OpenAIRE graph
Found an issue? Give us feedback
selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
3
Average
Top 10%
Average
Related to Research communities
Upload OA version
Are you the author of this publication? Upload your Open Access version to Zenodo!
It’s fast and easy, just two clicks!