Concepts of Dimension for Convex Geometries
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913189204066304 |
|---|---|
| author | Knauer, Kolja Trotter, William T. |
| author_facet | Knauer, Kolja Trotter, William T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08945 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Concepts of Dimension for Convex Geometries Knauer, Kolja Trotter, William T. Combinatorics Discrete Mathematics 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. |
| title | Concepts of Dimension for Convex Geometries |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2303.08945 |