Dualizing sup-preserving endomaps of a complete lattice

TL;DR

This paper explores the deep connections between dualizing endomaps, complete distributivity, and Girard quantale structures in the context of complete lattices, extending previous results with new insights into duality and automorphisms.

Contribution

It introduces a novel link between dualizing elements of [L,L], automorphisms of L, and the structure of complete distributivity, expanding the theory of quantale and quantaloid structures.

Findings

[L,L] has a Frobenius quantale structure iff L is completely distributive.

Dualizing elements of [L,L] correspond to automorphisms of L.

If L is finite and [L,L] is autodual, then L is distributive.

Abstract

It is argued in (Eklund et al., 2018) that the quantale [L,L] of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in (Santocanale, 2020) that this Girard quantale structure arises from the dual quantale of inf-preserving endomaps of L via Raney's transforms and extends to a Girard quantaloid structure on the full subcategory of SLatt (the category of complete lattices and sup-preserving maps) whose objects are the completely distributive lattices. It is the goal of this talk to illustrate further this connection between the quantale structure, Raney's transforms, and complete distributivity. Raney's transforms are indeed mix maps in the isomix category SLatt and most of the theory can be developed relying on naturality of these maps. We complete then the remarks on cyclic elements of [L,L] developed in (Santocanale, 2020) by investigating its dualizing elements. We argue that if [L,L] has the structure a Frobenius quantale, that is, if it has a dualizing element, not necessarily a cyclic one, then L is once more completely distributive. It follows then from a general statement on involutive residuated lattices that there is a bijection between dualizing elements of [L,L] and automorphisms of L. Finally, we also argue that if L is finite and [L,L] is autodual, then L is distributive.