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A point-free perspective on lax extensions and predicate liftings

Published online by Cambridge University Press:  01 December 2023

Sergey Goncharov ,
Dirk Hofmann Open the ORCID record for Dirk Hofmann [Opens in a new window] ,
Pedro Nora ,
Lutz Schröder  and
Paul Wild
Sergey Goncharov
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany
Dirk Hofmann*
Affiliation:
Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal
Pedro Nora
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany
Lutz Schröder
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany
Paul Wild
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany
*
Corresponding author: Dirk Hofmann; Email: dirk@ua.pt
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Abstract

Lax extensions of set functors play a key role in various areas, including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between lax extensions and predicate liftings from the point of view of quantale-enriched relations. Using this perspective, we show in particular that various fundamental concepts and results arise naturally and their proofs become very elementary. Ultimately, we prove that every lax extension is induced by a class of predicate liftings; we discuss several implications of this result.

Keywords

Lax extensioncoalgebraic modal logicKantorovich extensionMoss lifting

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 34 , Issue 2 , February 2024 , pp. 98 - 127
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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