Congruences of fork extensions of slim, planar, semimodular lattices
نویسندگان
چکیده
منابع مشابه
Notes on Planar Semimodular Lattices. Ii. Congruences
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A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim lattices are planar. Slim semimodular lattices play the main role in [3], where lattice theory is applied to a purely group theoretical problem. After exploring some easy properties of slim lattices and slim semimodular lattices, we give two visual structure theorems for slim semimodular lattices.
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A recent result of G. Czédli and E. T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding “forks”. We give a construction that accomplishes the same by deleting elements, by “resections”.
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A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ 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...
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We prove that every finite lattice has a congruence-preserving extension to a finite semimodular lattice.
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ژورنال
عنوان ژورنال: Algebra universalis
سال: 2016
ISSN: 0002-5240,1420-8911
DOI: 10.1007/s00012-016-0394-z