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ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

Published online by Cambridge University Press:  12 August 2016

NADAV MEIR
NADAV MEIR*
Affiliation:
DEPARTMENT OF MATHEMATICS BEN-GURION UNIVERSITY OF THE NEGEV P.O.B 653, BE’ER SHEVA 8410501, ISRAELE-mail: mein@math.bgu.ac.ilURL: http://www.math.bgu.ac.il/∼mein
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Abstract

We say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

Keywords

03C1003C3505D1005C55indivisibilityelementary indivisibilitycoloringquantifier elimination
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 951 - 971
Copyright
Copyright © The Association for Symbolic Logic 2016 

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