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A Gelfand duality for continuous lattices

Ruiyuan Chen

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 $Φ$, dual to a class of meets" for which "$Φ$-continuous lattice" and "$Φ$-algebraic lattice" are different notions, thus for which a $2$-valued duality does not suffice.

A Gelfand duality for continuous lattices

Abstract

We prove that the category of continuous lattices and meet- and directed join-preserving maps is dually equivalent, via the hom functor to , to the category of complete Archimedean meet-semilattices equipped with a finite meet-preserving action of the monoid of continuous monotone maps of fixing . 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 , dual to a class of meets" for which "-continuous lattice" and "-algebraic lattice" are different notions, thus for which a -valued duality does not suffice.
Paper Structure (5 sections, 27 theorems, 40 equations)

This paper contains 5 sections, 27 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. $\Phi$-continuous lattices
  3. Commuting meets and joins
  4. $\#U$-posets
  5. The duality

Key Result

theorem 1

Hom into $\#I$ yields a dual equivalence of categories between continuous lattices and complete $\^{\#U}$-modules.

Theorems & Definitions (61)

  • theorem 1: \ref{['thm:ctslat-dual-umod']}
  • theorem 2: \ref{['thm:cdlat-dual']}
  • theorem 3: \ref{['thm:sound-omega']}
  • definition 1
  • remark 1
  • definition 2
  • proposition 1
  • proof
  • proposition 2
  • proof
  • ...and 51 more