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A METRIC VERSION OF SCHLICHTING’S THEOREM

Part of: Algebraic combinatorics Connections with logic Structure and classification of infinite or finite groups Algebraic logic Ordered sets Semigroups

Published online by Cambridge University Press:  07 September 2020

ITAÏ BEN YAACOV  and
FRANK O. WAGNER
ITAÏ BEN YAACOV
Affiliation:
UNIVERSITÉ DE LYON, CNRS UNIVERSITÉ CLAUDE BERNARD LYON 1 INSTITUT CAMILLE JORDAN UMR5208 43 BD DU 11 NOVEMBRE 1918 69622 VILLEURBANNE CEDEX, FRANCEE-mail: benyaacov@math.univ-lyon1.frE-mail: wagner@math.univ-lyon1.frURL: http://math.univ-lyon1.fr/~wagner/URL: http://math.univ-lyon1.fr/~begnac/
FRANK O. WAGNER
Affiliation:
UNIVERSITÉ DE LYON, CNRS UNIVERSITÉ CLAUDE BERNARD LYON 1 INSTITUT CAMILLE JORDAN UMR5208 43 BD DU 11 NOVEMBRE 1918 69622 VILLEURBANNE CEDEX, FRANCEE-mail: benyaacov@math.univ-lyon1.frE-mail: wagner@math.univ-lyon1.frURL: http://math.univ-lyon1.fr/~wagner/URL: http://math.univ-lyon1.fr/~begnac/
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Abstract

If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.

Keywords

Schlichting’s Theoremuniformly commensurableclose-knit familycontinuous logicinvariant

MSC classification

Primary: 03G10: Lattices and related structures
Secondary: 05E15: Combinatorial aspects of groups and algebras 20M10: General structure theory 06A12: Semilattices 12L12: Model theory 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1607 - 1613
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Copyright
© The Association for Symbolic Logic 2020

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References

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