A Duality Between Hypergraphs and Cone Lattices
<p> 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 <i>L</i> is order isomorphic to the fram...
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| Language: | EN |
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Middle Tennessee State University
2018
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| Online Access: | http://pqdtopen.proquest.com/#viewpdf?dispub=10784141 |
| Summary: | <p> 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 <i>L</i> is order isomorphic to the frame of opens for a hypergraph endowed with the Classical topology. 2. A lattice <i> L</i> is bialgebraic, distributive, and its subposet of completely joinprime elements forms the incidence poset for a hypergraph. 3. A lattice <i> L</i> is a cone lattice. </p><p> 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. </p><p> |
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