## Algebraic and topological properties of finite partially ordered sets |

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abelian algebraic topology ascending subset barycentric subdivision called Chapter claim classification problems closed simplex Cohen-Macaulay cohomology group combinatorial compute Cond(S continuous map contractible carrier Corollary define definition denote Ep,q EP'q exists fibers finite posets fixed point property flasque diagram functor Galois connection geometric realization graded K-algebra graded poset greatest element Hence homology homotopy equivalence homotopy type ideal relation induction injective resolution isomorphic isotone map least element Lemma Let f lexicographic shellable map f map h maximal chain maximal elements Mobius algebra Mobius function Mobius number module monotone maps morphism nonempty chains obtain Obviously order complex A(P Paragraph partially ordered sets Proof Proposition prove R-algebra homomorphism R-modules rank ring Rota's theorem sd(P semi Galois map semimodular set of coatoms sh(A shelling simplicial complex spectral sequence subposet supersolvable topological space x,y e y e Q