Elementary submodels of parametrizable models

We introduce the notion of $F$-parametrizable model and prove some general results on elementary submodels of $F$-parametrizable models. Using this notion, we can uniformly characterize all elementary submodels for the field of real numbers and for the group of all permutations on natural numbers in the first order language as well as in the language of hereditarily finite superstructures. Assuming the constructibility axiom, we obtain a simpler characterization of elementary submodels of $F$-parametrizable models and prove some additional properties of the structure of their elementary submodels.