Geodesic
Nonexistence of Nonmolecular Generic Sets
Publications de l'Institut Mathématique, _N_S_36 (1984) no. 50, p. 29 .

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Generic subsets of partially ordered sets play an important role in giving significant examples of Zermelo-Fraenkel set-theoretical models. The significance of these models lies in the fact that a generic subset $G$ of a partially ordered set $P$, in general, does not exist in a model $M$ in which $P$ exists. Thus, by adjoining $G$ to $M$ an interesting extended model may ensue which has properties not shared by $M$. Thus, in considering generic extensions of set-theoretical models it is quite relevant to know whether or not a generic subset of a partially ordered set $P$ exists in the same model in which $P$ exists. In this paper, we give a necessary and sufficient condition for $P$ to have a generic subset in the same model.
Classification : 06A10 
@article{PIM_1984_N_S_36_50_a4,
     author = {Donald D. Steiner and Alexander Abian},
     title = {Nonexistence of {Nonmolecular} {Generic} {Sets}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {29 },
     publisher = {mathdoc},
     volume = {_N_S_36},
     number = {50},
     year = {1984},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1984_N_S_36_50_a4/}
}
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Donald D. Steiner; Alexander Abian. Nonexistence of Nonmolecular Generic Sets. Publications de l'Institut Mathématique, _N_S_36 (1984) no. 50, p. 29 . http://geodesic.mathdoc.fr/item/PIM_1984_N_S_36_50_a4/