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Topological representation of lattice homomorphisms

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

Wallman proved that if $mathbb{L}$ is a distributive lattice with $mathbf{0}$ and $mathbf{1}$, then there is a $T_1$-space with a base (for closed subsets) being a homomorphic image of $mathbb{L}$. We show that this theorem can be extended over homomorphisms. More precisely: if $f{Lat}$ denotes the category of normal and distributive lattices with $mathbf{0}$ and $mathbf{1}$ and homomorphisms, and $f{Comp}$ denotes the category of compact Hausdorff spaces and continuous mappings, then there exists a contravariant functor $mathcal{W}:f{Lat} of{Comp}$. When restricted to the subcategory of Boolean lattices this functor coincides with a well-known Stone functor which realizes the Stone Duality. The functor $mathcal{W}$ carries monomorphisms into surjections. However, it does not carry epimorphisms into injections. The last property makes a difference with the Stone functor. Some applications to topological constructions are given as well.

This talk is part of the Isaac Newton Institute Seminar Series series.

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