skip to content

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.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons