Geodesic
Embedding partially ordered sets into $^ω ω$
Fundamenta Mathematicae, Tome 151 (1996) no. 1, pp. 53-95

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We investigate some natural questions about the class of posets which can be embedded into 〈ω,≤*〉. Our main tool is a simple ccc forcing notion $H_E$ which generically embeds a given poset E into 〈ω,≤*〉 and does this in a "minimal" way (see Theorems 9.1, 10.1, 6.1 and 9.2). 
Ilijas  Farah. Embedding partially ordered sets into $^ω ω$. Fundamenta Mathematicae, Tome 151 (1996) no. 1, pp. 53-95. doi: 10.4064/fm-151-1-53-95
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