Unitless Frobenius quantales

TL;DR

This paper introduces and develops the theory of non-unital Frobenius quantales using negation as a primitive, providing new representations, examples, and showing the impossibility of adding units without losing negation.

Contribution

It defines non-unital Frobenius quantales with negation, develops their elementary theory, and demonstrates their properties and examples, including phase semantics and representation theorems.

Findings

Tight endomaps of complete lattices form unital Girard quantales iff the lattice is completely distributive.

Frobenius structures are exemplified by trace class operators on infinite-dimensional Hilbert spaces.

Adding units to Frobenius quantales cannot preserve negation, showing units cannot be properly incorporated.

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 Mn 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.