Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T21:07:46.531Z Has data issue: false hasContentIssue false

An introduction to b-minimality

Published online by Cambridge University Press:  28 January 2010

S. Barry Cooper
Affiliation:
University of Leeds
Herman Geuvers
Affiliation:
Radboud Universiteit Nijmegen
Anand Pillay
Affiliation:
University of Leeds
Jouko Väänänen
Affiliation:
University of Amsterdam and University of Helsinki
Get access

Summary

Abstract. We give a survey with some explanations but no proofs of the new notion of b-minimality by the author and F. Loeser [b-minimality, Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 195–227, math.LO/0610183]. We compare this notion with other notions like o-minimality, C-minimality, p-minimality, and so on.

Introduction. As van den Dries notes in his book [17], Grothendieck's dream of tame geometries found a certain realization in model theory, at first by the study of the geometric properties of definable sets for some nice structure like the field of real numbers, and then by axiomatizing these properties by notions of o-minimality, minimality, C-minimality, p-minimality, v-minimality, t-minimality, b-minimality, and so on. Although there is a joke speaking of x-minimality with x = a, b, c, d, …, these notions are useful and needed in different contexts for different kinds of structures, for example, o-minimality is for ordered structures, and v-minimality is for algebraically closed valued fields.

In recent work with F. Loeser [4], we tried to unify some of the notions of x-minimality for different x, for certain x only under extra conditions, to a very basic notion of b-minimality. At the same time, we tried to keep this notion very flexible, very tame with many nice properties, and able to describe complicated behavior.

An observation of Grothendieck's is that instead of looking at objects, it is often better to look at morphisms and study the fibers of the morphisms.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×