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Convex geometries representable by at most 5 circles on the plane

PolyMath REU Convex Geometries Collaboration, Kira Adaricheva, Madina Bolat, Gent Gjonbalaj, Brandon Amerine, J. Alexandria Behne, Evan Daisy, Alexander Frederiksen, Ayush Garg, Zachary King, Grace Ma, Michelle Olson, Rohit Pai, Junewoo Park, Cat Raanes, Sean Riedel, Joseph Rogge, Raviv Sarch, James Thompson, Fernanda Yepez-Lopez, Stephanie Zhou

Abstract

A convex geometry is a closure system satisfying the anti-exchange property. In this work we document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 non-isomorphic geometries on a 4-element set can be represented by circles, and of the 672 geometries on a 5-element set, we made representations of 623. Of the 49 remaining geometries on a 5-element set, one was already shown not to be representable due to the Weak Carousel property, as articulated by Adaricheva and Bolat (Discrete Mathematics, 2019). In this paper we show that 7 more of these convex geometries cannot be represented by circles on the plane, due to what we term the Triangle Property.

Convex geometries representable by at most 5 circles on the plane

Abstract

A convex geometry is a closure system satisfying the anti-exchange property. In this work we document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 non-isomorphic geometries on a 4-element set can be represented by circles, and of the 672 geometries on a 5-element set, we made representations of 623. Of the 49 remaining geometries on a 5-element set, one was already shown not to be representable due to the Weak Carousel property, as articulated by Adaricheva and Bolat (Discrete Mathematics, 2019). In this paper we show that 7 more of these convex geometries cannot be represented by circles on the plane, due to what we term the Triangle Property.

Paper Structure

This paper contains 20 sections, 13 theorems, 10 equations, 666 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Definitions
  4. Convex Dimension
  5. Convex hull operator
  6. Creating the list of geometries on 4- and 5-element sets
  7. Representing the list
  8. Representation of geometries on 5-element set based on representations on 4-element or 3-element set
  9. Checking correctness of representations
  10. GeoGebra as a toolkit for representations
  11. Google Colab toolkit
  12. Triangle Property and Impossible geometries
  13. Tight implication on circles
  14. Triangle Property
  15. Acknowledgments.
  16. ...and 5 more sections

Key Result

Theorem \oldthetheorem

Every alignment can be expressed as the join of some collection of monotone alignments on the same base set.

Figures (666)

  • Figure 1: Example of a set which is convex and a set which is not convex.
  • Figure 1: Boolean
  • Figure 2: The convex hull $\mathop{\mathrm{CH}}\nolimits(ABCDEFG)$
  • Figure 2: $abcd \to e$
  • Figure 3: Examples of convex hulls
  • ...and 661 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem \oldthetheorem
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 29 more