A Gelfand duality for continuous lattices

Ruiyuan Chen

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arXiv · 2301.05988

A Gelfand duality for continuous lattices

Ruiyuan Chen

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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.

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BibTeX RIS

Ruiyuan Chen. 2023-01-14. A Gelfand duality for continuous lattices. https://arxiv.org/abs/2301.05988

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