We use model theoretic forcing to study and generalize the construction of ($K, \leq$)-generic models introduced by Kueker and Laskowski. We characterize the ($K, \leq$)-generic models in terms of forcing and introduce a more general class of models, called essential forcing generics, which have many of the same properties.
"Essential Forcing Generics." Notre Dame J. Formal Logic 41 (1) 41 - 52, 2000. https://doi.org/10.1305/ndjfl/1027953482