Frankl’s conjecture for large semimodular and planar semimodular lattices

Czédli, Gábor, and Schmidt, E. Tamás. "Frankl’s conjecture for large semimodular and planar semimodular lattices." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 47.1 (2008): 47-53. <http://eudml.org/doc/32473>.

@article{Czédli2008,
abstract = {A lattice $L$ is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element $f\in L$ such that at most half of the elements $x$ of $L$ satisfy $f\le x$. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let $m$ denote the number of nonzero join-irreducible elements of $L$. It is well-known that $L$ consists of at most $2^m$ elements. Let us say that $L$ is large if it has more than $5\cdot 2^\{m-3\}$ elements. It is shown that every large semimodular lattice satisfies Frankl’s conjecture. The second result states that every finite semimodular planar lattice $L$ satisfies Frankl’s conjecture. If, in addition, $L$ has at least four elements and its largest element is join-reducible then there are at least two choices for the above-mentioned $f$.},
author = {Czédli, Gábor, Schmidt, E. Tamás},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {union-closed sets; Frankl’s conjecture; lattice; semimodularity; planar lattice; Frankl's conjecture; union-closed sets conjecture; finite lattice; join-irreducible elementsfinit semimodular planar lattices; semimodularity},
language = {eng},
number = {1},
pages = {47-53},
publisher = {Palacký University Olomouc},
title = {Frankl’s conjecture for large semimodular and planar semimodular lattices},
url = {http://eudml.org/doc/32473},
volume = {47},
year = {2008},
}

TY - JOUR
AU - Czédli, Gábor
AU - Schmidt, E. Tamás
TI - Frankl’s conjecture for large semimodular and planar semimodular lattices
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2008
PB - Palacký University Olomouc
VL - 47
IS - 1
SP - 47
EP - 53
AB - A lattice $L$ is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element $f\in L$ such that at most half of the elements $x$ of $L$ satisfy $f\le x$. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let $m$ denote the number of nonzero join-irreducible elements of $L$. It is well-known that $L$ consists of at most $2^m$ elements. Let us say that $L$ is large if it has more than $5\cdot 2^{m-3}$ elements. It is shown that every large semimodular lattice satisfies Frankl’s conjecture. The second result states that every finite semimodular planar lattice $L$ satisfies Frankl’s conjecture. If, in addition, $L$ has at least four elements and its largest element is join-reducible then there are at least two choices for the above-mentioned $f$.
LA - eng
KW - union-closed sets; Frankl’s conjecture; lattice; semimodularity; planar lattice; Frankl's conjecture; union-closed sets conjecture; finite lattice; join-irreducible elementsfinit semimodular planar lattices; semimodularity
UR - http://eudml.org/doc/32473
ER -