Unitless Frobenius quantales

by   Cédric de Lacroix, et al.

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.


page 1

page 2

page 3

page 4


Frobenius structures in star-autonomous categories

It is known that the quantale of sup-preserving maps from a complete lat...

Dualizing sup-preserving endomaps of a complete lattice

It is argued in (Eklund et al., 2018) that the quantale [L,L] of sup-pre...

Adaptive directional Haar tight framelets on bounded domains for digraph signal representations

Based on hierarchical partitions, we provide the construction of Haar-ty...

Revisiting Semantics of Interactions for Trace Validity Analysis

Interaction languages such as UML sequence diagrams are often associated...

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce th...

Formalized functional analysis with semilinear maps

Semilinear maps are a generalization of linear maps between vector space...

Predicative Aspects of Order Theory in Univalent Foundations

We investigate predicative aspects of order theory in constructive univa...