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Generalizing the proof of the theorem describing the closed cone of flag f-vectors of arbitrary graded posets, we give a description of the cone of flag f-vectors of planar graded posets. The labeling used is a special case of a “chain-edge labeling with the first atom property”, or FA-labeling, which also generalizes the notion of lexicographic shelling, or CL-labeling. The resulting analogy suggests...
There is a natural way to associate with a poset P a hypergraph HP, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of HP. The ordinary graph GP of incomparable pairs determined by the edges in HP of size 2 can have chromatic number substantially less than HP. We give a new proof of the fact that the dimension of P is 2 if and only if GP is bipartite...
The recognition complexity of ordered set properties is considered in terms of how many questions must be put to an adversary to decide if an unknown partial order has the prescribed property. We prove a lower bound of order n2 for properties that are characterized by forbidden substructures of fixed size. For the properties being connected, and having exactly k comparable pairs, k ≤ n2 / 4 we show...
We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum...
We study a visibility relation on the nonempty connected convex subsets of a finite partially ordered set and we investigate the partial orders representable as a visibility relation of such subsets of a weak order. Moreover, we consider restrictions where the subsets of the weak order are total orders or isomorphic total orders.
O*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F*-rings. We show that the two classes of rings coincide.
It is proved that the collection of all finite lattices with the same partially ordered set of meet-irreducible elements can be ordered in a natural way so that the obtained poset is a lattice. Necessary and sufficient conditions under which this lattice is Boolean, distributive and modular are given.
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