Structure of k-CS- Transitive Cycle-Free Partial Orders.

Intro -- Contents -- 1 Extended Introduction -- 1.1 Introduction -- 1.2 Cycle-free partial orders -- 1.3 Homogeneous structures -- 1.4 k-connected set transitivity -- 1.5 Finite and infinite chain CFPO[sub(s)] -- 1.6 Elements of the classification -- 1.7 Further work -- 2 Preliminaries -- 2.1 Introduction -- 2.2 Dedekind-complete partial orders -- 2.3 Cycle-free partial orders -- 2.4 Concerning paths, and the density lemma -- 2.5 Substructures, cones, and their extensions -- 2.6 Convex cycle-free partial orders -- 3 Properties of k-CS-transitive CFPOs -- 3.1 Introduction -- 3.2 k-CS-transivity and k-CS-homogeneity -- 3.3 The infinite chain case -- 3.4 The finite chain case and the bipartite theorem -- 3.5 Sporadic and skeletal cycle-free partial orders -- 4 Constructing CFPOs -- 4.1 Introduction -- 4.2 The completion theorem (Part one) -- 4.3 The completion theorem (Part two) -- 4.4 Useful results concerning M,M[sup(D)] and M -- 5 Characterization and Isomorphism Theorems -- 5.1 Introduction -- 5.2 Characterizations in the infinite chain case -- 5.3 The isomorphism theorems and their corollaries -- 6 Classification of skeletal CFPOs (Part 1) -- 6.1 Introduction -- 6.2 Case A: ↑Ram(M) = ↓Ram(M) -- 6.3 Case B: ↑Ram(M)∩↓Ram(M) = φand Ram(M) is dense -- 6.4 Covering orders -- 6.5 Case C: Fully covered cycle-free partial orders -- 6.6 Case D: Partially covered cycle-free partial orders -- 6.7 Subcase D1: The cycle-free partial orders D[sup(d,u,u')][sub(σ)] -- 6.8 Subcase D2: The cycle-free partial orders e[sup(d,u,u')][sub(σ)] -- 6.9 Subcase D3: The cycle-free partial orders F[sup(d,u,u')][sub(σ,z)] -- 6.10 Summary -- 7 Classification of skeletal CFPOs (Part 2) -- 7.1 Introduction -- 7.2 The cycle-free partial orders g[sup(u,d,u',d')][sub(z)] -- 7.3 Case 2: The cycle-free partial orders H[sup(u,d,u',d')][sub(z)] -- 7.4 An empty case.

7.5 Case 3: The remaining few -- 7.6 Conclusions in the skeletal case -- Appendix: Sporadic Cycle-free Partial Orders -- A.1 Introduction -- A.2 The classification -- A.3 Conclusions.

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