Contents

Extended Introduction 1

1.1 Introduction 1

1.2 Cycle-free partial orders 2

1.3 Homogeneous structures 7

1.4 fc-connected set transitivity 10

1.5 Finite and infinite chain CFPOs 13

1.6 Elements of the classification 17

1.7 Further work 24

Preliminaries 25

2.1 Introduction 25

2.2 Dedekind-complete partial orders 26

2.3 Cycle-free partial orders 28

2.4 Concerning paths, and the density lemma 34

2.5 Substructures, cones, and their extensions 38

x cycle-free partial orders 41

Properties of k-CS-transitive CFPO s 43

3.1 Introduction 43

3.2 ifc-CS-transivity and fc-CS-homogeneity 44

3.3 The infinite chain case 45

3.4 The finite chain case and the bipartite theorem 52

3.5 Sporadic and skeletal cycle-free partial orders 53

Constructing CFPOs 64

4.1 Introduction 64

4.2 The completion theorem (Part one) 72

4.3 The completion theorem (Part two) _ 80

4.4 Useful results concerning M,MD and M 85

Characterization and Isomorphism Theorems 92

5.1 Introduction 92

5.2 Characterizations in the infinite chain case 93

5.3 The isomorphism theorems and their corollaries 102

Classification of skeletal CFPOs (Part 1) 112

6.1 Introduction 112

6.2 Case A: |Ram(M) = jRam(M) 112

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