The Involutive Quantaloid of Completely Distributive Lattices

TL;DR

This paper characterizes when the quantale of join-continuous endo-functions on a complete lattice is involutive, showing it occurs precisely for completely distributive lattices, and explores the categorical structure of such lattices.

Contribution

It establishes a necessary and sufficient condition for the quantale to be involutive, linking it to complete distributivity of the lattice, and analyzes the categorical properties of these lattices.

Findings

Q(L) is involutive iff L is completely distributive.

The dual tensor operation corresponds to composition in the dual quantale.

The category of completely distributive lattices forms an involutive quantaloid.

Abstract

Let L be a complete lattice and let Q(L) be the unital quantale of join-continuous endo-functions of L. We prove the following result: Q(L) is an involutive (that is, non-commutative cyclic $\star$-autonomous) quantale if and only if L is a completely distributive lattice. If this is the case, then the dual tensor operation corresponds, via Raney's transforms, to composition in the (dual) quantale of meet-continuous endo-functions of L. Let sLatt be the category of sup-lattices and join-continuous functions and let cdLatt be the full subcategory of sLatt whose objects are the completely distributive lattices. We argue that (i) cdLatt is itself an involutive quantaloid, and therefore it is the largest full-subcategory of sLatt with this property; (ii) cdLatt is closed under the monoidal operations of sLatt and, consequently, if Q(L) is involutive, then Q(L) is completely distributive as well.