Quasiplanar Diagrams and Slim Semimodular Lattices
نویسندگان
چکیده
منابع مشابه
Slim Semimodular Lattices. I. A Visual Approach
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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In a ranked lattice, we consider two maximal chains, or “flags” to be i-adjacent if they are equal except possibly on rank i . Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a “Jordan-Hölder permutation” between any two flags. This permutation has the properties of an Sn-distance function on the chamber system of flags. Using these noti...
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Let ~ H and ~ K be finite composition series of a group G. The intersections Hi ∩ Kj of their members form a lattice CSL( ~ H, ~ K) under set inclusion. Improving the Jordan-Hölder theorem, G. Grätzer, J.B. Nation and the present authors have recently shown that ~ H and ~ K determine a unique permutation π such that, for all i, the i-th factor of ~ H is “down-and-up projective” to the π(i)-th f...
متن کاملThe Matrix of a Slim Semimodular Lattice
A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim semimodular lattices play the main role in G. Czédli and E.T. Schmidt [5], where lattice theory is applied to a purely group theoretical problem. Here we develop a unique matrix representation for these lattices.
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ژورنال
عنوان ژورنال: Order
سال: 2015
ISSN: 0167-8094,1572-9273
DOI: 10.1007/s11083-015-9362-z