Girard couples of quantales

TL;DR

This paper introduces Girard couples, a new duality-based structure linking two quantales, with applications in endomorphism quantales and operator algebra spectra, and constructs examples from arbitrary sup-lattices.

Contribution

It defines Girard couples, explores their properties, and provides a construction for Girard quantales from any sup-lattice, expanding the understanding of duality in quantale theory.

Findings

Girard couples generalize duality concepts in quantales.

Constructed Girard quantale from any sup-lattice.

Applications in operator algebra spectra and endomorphism quantales.

Abstract

We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice $S$, a Girard quantale whose right-sided part is isomorphic to $S$.