Convex Geometries yielded by Transit Functions
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913375224594432 |
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| author | Changat, Manoj Sheela, Lekshmi Kamal K. Peterin, Iztok Shanavas, Ameera Vaheeda |
| author_facet | Changat, Manoj Sheela, Lekshmi Kamal K. Peterin, Iztok Shanavas, Ameera Vaheeda |
| contents | Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$ hold for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset K$ for every $u,v\in K$ and all $R$-convex subsets of $V$ form a convexity $\mathcal{C}_R$. We consider Minkowski-Krein-Milman property that every $R$-convex set $K$ in a convexity $\mathcal{C}_R$ is the convex hull of the set of extreme points of $K$ from axiomatic point of view and present a characterization of it. Later we consider several well-known transit functions on graphs and present the use of the mentioned characterizations on them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01100 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convex Geometries yielded by Transit Functions Changat, Manoj Sheela, Lekshmi Kamal K. Peterin, Iztok Shanavas, Ameera Vaheeda Combinatorics 52A01, 05C38 Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$ hold for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset K$ for every $u,v\in K$ and all $R$-convex subsets of $V$ form a convexity $\mathcal{C}_R$. We consider Minkowski-Krein-Milman property that every $R$-convex set $K$ in a convexity $\mathcal{C}_R$ is the convex hull of the set of extreme points of $K$ from axiomatic point of view and present a characterization of it. Later we consider several well-known transit functions on graphs and present the use of the mentioned characterizations on them. |
| title | Convex Geometries yielded by Transit Functions |
| topic | Combinatorics 52A01, 05C38 |
| url | https://arxiv.org/abs/2406.01100 |