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Topological representation of lattice homomorphisms

Presented by: 
A Blaszczyk University of Silesia in Katowice
Monday 24th August 2015 - 14:00 to 14:30
INI Seminar Room 1
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 $\bf{Lat}$ denotes the category of normal and distributive lattices with $\mathbf{0}$ and $\mathbf{1}$ and homomorphisms, and $\bf{Comp}$ denotes the category of compact Hausdorff spaces and continuous mappings, then there exists a contravariant functor $\mathcal{W}:\bf{Lat}\to\bf{Comp}$. 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.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons