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Unitless Frobenius Quantales

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Abstract

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices \(M_n\) and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.

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Acknowledgements

The authors are thankful to Nick Galatos, the anonymous referee, and the editor for precious pointers, remarks, and guidance for improving a first version of this paper.

Funding

This work was supported by the Agence Nationale de la Recherche, Project LAMBDACOMB ANR-21-CE48-0017.

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Author notes

  1. Cédric de Lacroix and Luigi Santocanaley have contributed equally to this work.

Authors and Affiliations

  1. LIS, CNRS UMR 7020, Aix-Marseille Université, Marseille, France

    Cédric de Lacroix & Luigi Santocanale

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  1. Cédric de Lacroix
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  2. Luigi Santocanale
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Correspondence to Luigi Santocanale.

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Communicated by Jiří Rosický.

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de Lacroix, C., Santocanale, L. Unitless Frobenius Quantales. Appl Categor Struct 31, 5 (2023). https://doi.org/10.1007/s10485-022-09699-5

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