242 research outputs found
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
A class of infinite convex geometries
Various characterizations of finite convex geometries are well known. This
note provides similar characterizations for possibly infinite convex geometries
whose lattice of closed sets is strongly coatomic and lower continuous. Some
classes of examples of such convex geometries are given.Comment: 10 page
Notes on the description of join-distributive lattices by permutations
Let L be a join-distributive lattice with length n and width(Ji L) \leq k.
There are two ways to describe L by k-1 permutations acting on an n-element
set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a
recent lattice theoretical way of the second author. We prove that these two
approaches are equivalent. Also, we characterize join-distributive lattices by
trajectories.Comment: 8 pages, 1 figur
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