A Duality between hypergraphs and cone lattices
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Middle Tennessee State University
Abstract
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.
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.