Concepts of Dimension for Convex Geometries

TL;DR

The paper systematically analyzes six dimension notions for convex geometries—$ ext{cdim}$, $ ext{vcdim}$, $ ext{dim}$, $ ext{bdim}$, $ ext{ldim}$, and $ ext{fdim}$—along with the standard example number $ ext{se}$, establishing a landscape of inequalities and separations among them. It proves a striking separation: there exist convex geometries with $ ext{dim}(P)=3$ yet $ ext{cdim}(P)=n+1$ for infinitely many $n$, showing $ ext{dim}$ does not bound $ ext{cdim}$. It also shows that when $ ext{dim}(P)\le 2$, then $ ext{cdim}(P)= ext{dim}(P)$, and it clarifies the relation between $ ext{se}$ and $ ext{maxdd}$, proving $ ext{se}(P)= ext{maxdd}(P)$ for convex geometries. To illustrate separations beyond VC-dimension, the authors construct the family $P(k,n)$ where $ ext{bdim}$ and $ ext{ldim}$ grow with $n$ (for fixed $k$), while $ ext{cdim}$, $ ext{dim}$, and $ ext{se}$ remain controlled and $ ext{fdim}$ stays below $2^{k+1}$. The work concludes with open problems and directions for future study of dimensions in convex geometries.

Abstract

Let $X$ be a finite set. A family $P$ of subsets of $X$ is called a convex geometry with ground set $X$ if (1) $\emptyset, X\in P$; (2) $A\cap B\in P$ whenever $A,B\in P$; and (3) if $A\in P$ and $A\neq X$, there is an element $α\in X-A$ such that $A\cup\{α\}\in P$. As a non-empty family of sets, a convex geometry has a well defined VC-dimension. In the literature, a second parameter, called convex dimension, has been defined expressly for these structures. Partially ordered by inclusion, a convex geometry is also a poset, and four additional dimension parameters have been defined for this larger class, called Dushnik-Miller dimension, Boolean dimension, local dimension, and fractional dimension, espectively. For each pair of these six dimension parameters, we investigate whether there is an infinite class of convex geometries on which one parameter is bounded and the other is not.

Figures (3)

• Figure 1: For $n\ge3$, we illustrate a $3$-dimensional poset $Q_n$ and its subposet $P_n$ obtained by removing the grey points. Then $P_n$ is a convex geometry, and the white points constitute the set of meet-irreducible elements of $P_n$.
• Figure 2: On the left, we show a convex geometry with a planar order diagram. Note that maximum down degree is $2$, and every interior face is a diamond. On the right, we suggest how an element in the interior would appear if it had up degree $1$. We will show that this is impossible.
• Figure 3: We illustrate the convex geometry $P(1,5)$. In the drawing, sets are indicated without braces and commas. Although these facts will follow from the more general arguments given below, readers may enjoy verifying that $\operatorname{cdim}(P(1,5))=4$, $\dim(P(1,5))=\operatorname{bdim}(P(1,5))=\operatorname{fdim}(P(1,5))=3$, and $\operatorname{se}(P(1,5))= \operatorname{maxdd}(P(1,5))=2$.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 3.1: Boolean property
• Theorem 3.2
• Theorem 3.3
• proof
• Theorem 3.5