Two Forbidden Induced Minor Theorems for Antimatroids

TL;DR

This paper establishes two new forbidden induced minor theorems for antimatroids, viewed as proof systems, including a characterization of partial orders and a unique minimal proof system with conflicting dependencies.

Contribution

It introduces two novel forbidden induced minor theorems for antimatroids, expanding understanding of their structural properties as proof systems.

Findings

Characterization of partial orders as proof systems

Existence of a unique minimal proof system with conflicting dependencies

Extension of forbidden minor theorems to possibly infinite antimatroids

Abstract

Antimatroids were discovered by Dilworth in the context of lattices [4] and introduced by Edelman and Jamison as convex geometries in[5]. The author of the current paper independently discovered (possibly infinite) antimatroids in the context of proof systems in mathematical logic [1]. Carlson, a logician, makes implicit use of this view of proof systems as possibly infinite antimatroids in [2]. Though antimatroids are in a sense dual to matroids, far fewer antimatroid forbidden minor theorems are known. Some results of this form are proved in [6], [7], [8], and [9]. This paper proves two forbidden induced minor theorems for these objects, which we think of as proof systems. Our first main theorem gives a new proof of the forbidden induced minor characterization of partial orders as proof systems, proved in [8] in the finite case and stated in [10] for what we call strong aut descendable proof systems. It essentially states that, pathologies aside, there is a certain unique simplest nonposet. Our second main theorem states the new result that, pathologies aside, there is a certain unique simplest proof system containing points $x$ and $y$ such that $x$ needs $y$ in one context, yet $y$ needs $x$ in another.