The Involutive Quantaloid of Completely Distributive Lattices

Abstract

Let L be a complete lattice and let ${\mathcal {Q}}(L)$ be the unital quantale of join-continuous endo-functions of L. We prove that ${\mathcal {Q}}(L)$ has at most two cyclic elements, and that if it has a non-trivial cyclic element, then L is completely distributive and ${\mathcal {Q}}(L)$ is involutive (that is, non-commutative cyclic $\star$-autonomous). 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 $\mathsf {Latt}_{\bigvee }$ be the category of sup-lattices and join-continuous functions and let $\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}}$ be the full subcategory of $\mathsf {Latt}_{\bigvee }$ whose objects are the completely distributive lattices. We argue that $\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}}$ is itself an involutive quantaloid, thus it is the largest full-subcategory of $\mathsf {Latt}_{\bigvee }$ with this property. Since $\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}}$ is closed under the monoidal operations of $\mathsf {Latt}_{\bigvee }$, we also argue that if ${\mathcal {Q}}(L)$ is involutive, then ${\mathcal {Q}}(L)$ is completely distributive as well; consequently, any lattice embedding into an involutive quantale of the form ${\mathcal {Q}}(L)$ has, as its domain, a distributive lattice.