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On the Equimorphism Types of Linear Orderings

Published online by Cambridge University Press:  15 January 2014

Antonio Montalbán
Antonio Montalbán*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL, USAE-mail: antonio@math.uchicago.eduURL: www.math.uchicago.edu/~antonio
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§1. Introduction. A linear ordering (also known as total ordering) embeds into another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to be equimorphic if they can be embedded in each other. This is an equivalence relation, and we call the equivalence classes equimorphism types. We analyze the structure of equimorphism types of linear orderings, which is partially ordered by the embeddability relation. Our analysis is mainly fromthe viewpoints of Computability Theory and Reverse Mathematics. But we also obtain results, as the definition of equimorphism invariants for linear orderings, which provide a better understanding of the shape of this structure in general.

This study of linear orderings started by analyzing the proof-theoretic strength of a theorem due to Jullien [Jul69]. As is often the case in Reverse Mathematics, to solve this problem it was necessary to develop a deeper understanding of the objects involved. This led to a variety of results on the structure of linear orderings and the embeddability relation on them. These results can be divided into three groups.

Type
Communications
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 1 , March 2007 , pp. 71 - 99
Copyright
Copyright © Association for Symbolic Logic 2007

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