A Gelfand duality for continuous lattices

TL;DR

This paper establishes a duality between categories of continuous lattices and certain algebraic structures, extending classical Gelfand duality to a broader lattice-theoretic context.

Contribution

It introduces a new duality for continuous and completely distributive lattices, expanding the scope of Gelfand duality in lattice theory.

Findings

Duality between continuous lattices and complete Archimedean meet-semilattices.

Extension of duality to completely distributive lattices.

Characterization of well-behaved classes of joins and meets for duality.

Abstract

We prove that the category of continuous lattices and meet- and directed join-preserving maps is dually equivalent, via the hom functor to $[0,1]$, to the category of complete Archimedean meet-semilattices equipped with a finite meet-preserving action of the monoid of continuous monotone maps of $[0,1]$ fixing $1$. We also prove an analogous duality for completely distributive lattices. Moreover, we prove that these are essentially the only well-behaved "sound classes of joins $\Phi$, dual to a class of meets" for which "$\Phi$-continuous lattice" and "$\Phi$-algebraic lattice" are different notions, thus for which a $2$-valued duality does not suffice.