Distributive Lattices, Bipartite Graphs and Alexander Duality

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A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be constructed explicitly and a combinatorial formula to compute the Betti numbers of HP will be presented. Third, by using the fact that the Alexander dual of the simplicial complex Δ whose Stanley–Reisner ideal coincides with HP is Cohen–Macaulay, all the Cohen–Macaulay bipartite graphs will be classified.

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