A convex combinatorial property of compact sets in the plane and its roots in lattice theory | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 6، دوره 11، Special Issue Dedicated to Prof. George A. Grätzer، مهر 2019، صفحه 57-92 اصل مقاله (862.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.11.1.57 | ||
| نویسندگان | ||
| Gábor Czédli1؛ Árpád Kurusa2 | ||
| 1Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary | ||
| 2Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720 | ||
| چکیده | ||
| K. Adaricheva and M. Bolat have recently proved that if $\,\mathcal U_0$ and $\,\mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $\,\mathcal U_{1-k}$ is included in the convex hull of $\,\mathcal U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $\,\mathcal U_0$ and $\,\mathcal U_1$ are compact sets in the plane such that $\,\mathcal U_1$ is obtained from $\,\mathcal U_0$ by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Grätzer and E. Knapp, lead to our result. | ||
| کلیدواژهها | ||
| Congruence lattice؛ planar semimodular lattice؛ convex hull؛ compact set؛ linebreak circle؛ combinatorial geometry؛ abstract convex geometry؛ anti-exchange property | ||
| مراجع | ||
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