A Duality between hypergraphs and cone lattices
Zack French · ProQuest Demo Repository · 2018
In this paper, we introduce and characterize the class of lattices that arise as the family of lowersets of the incidence poset for a hypergraph. In particular, we show that the following statements are logically equivalent: 1. A lattice L is order isomorphic to the frame of opens for a hypergraph endowed with the Classical topology. 2. A lattice L is bialgebraic, distributive, and its subposet of completely joinprime elements forms the incidence poset for a hypergraph. 3. A lattice L is a cone lattice. We conclude the paper by extending a well-known Stone-type duality to the categories of hypergraphs coupled with finite-based HP-morphisms and cone lattices coupled with frame homomorphisms that preserve compact elements.