TY - JOUR
ID - 82639
TI - A convex combinatorial property of compact sets in the plane and its roots in lattice theory
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Czédli, Gábor
AU - Kurusa, Árpád
AD - Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary
AD - Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720
Y1 - 2019
PY - 2019
VL - 11
IS - Special Issue Dedicated to Prof. George A. Grätzer
SP - 57
EP - 92
KW - Congruence lattice
KW - planar semimodular lattice
KW - convex hull
KW - compact set
KW - linebreak circle
KW - combinatorial geometry
KW - abstract convex geometry
KW - anti-exchange property
DO - 10.29252/cgasa.11.1.57
N2 - 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.
UR - https://cgasa.sbu.ac.ir/article_82639.html
L1 - https://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf
ER -