The Mo bius number of a finite partially ordered set equals (up to sign) the difference between the number of even and odd edge covers of its incomparability graph. We use Alexander duality and the nerve lemma of algebraic topology to obtain a stronger result. It relates the homology of a finite simplicial complex 2 that is not a simplex to the cohomology of the complex 1 of nonempty sets of minimal non-faces that do not cover the vertex set of 2.1997 Academic Press
THE MO BIUS NUMBER RESULTRecall that if P =P _ [0 , 1 ] is a finite poset, the vertices of its incomparability graph G are the elements of P and the edges of G are the 2-element antichains in P. The Mo bius number + P (0 , 1 ) is, by Philip Hall's theorem, the reduced Euler characteristic /~(2) of the order complex
Poset-theoretic generalizations of set-theoretic committee constructions are
presented. The structure of the corresponding subposets is described. Sequences
of irreducible fractions associated to the principal order ideals of finite
bounded posets are considered and those related to the Boolean lattices are
explored; it is shown that such sequences inherit all the familiar properties
of the Farey sequences.Comment: 29 pages. Corrected version of original publication which is
available at http://www.springerlink.com, see Corrigendu
Abstract. Antichains of a finite bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers are discussed.
Abstract. Oriented matroids can serve as a tool of modeling of collective decision-making processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids. A tope committee for an oriented matroid is a subset of its maximal covectors such that every positive halfspace contains more than half of the covectors from this subset. For a large subfamily of oriented matroids their committee structure is quite rich; for example, any maximal chains in their tope posets provide one with information sufficient to construct a committee.
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