Geometry of antimatroidal point sets

TL;DR

This paper characterizes two-dimensional antimatroidal point sets as parallelogram polyominoes, revealing their convex dimension and exploring geometric properties of three-dimensional antimatroids closed under intersection.

Contribution

It provides a geometric characterization of 2D antimatroidal point sets and investigates properties of 3D antimatroids with intersection closure.

Findings

2D antimatroidal point sets are exactly parallelogram polyominoes

These sets have convex dimension 2

Properties of 3D antimatroids closed under intersection are analyzed

Abstract

The notion of "antimatroid with repetition" was conceived by Bjorner, Lovasz and Shor in 1991 as a multiset extension of the notion of antimatroid. When the underlying set consists of only two elements, such two-dimensional antimatroids correspond to point sets in the plane. In this research we concentrate on geometrical properties of antimatroidal point sets in the plane and prove that these sets are exactly parallelogram polyominoes. Our results imply that two-dimensional antimatroids have convex dimension 2. The second part of the research is devoted to geometrical properties of three-dimensional antimatroids closed under intersection.