In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length $\omega$ is sometimes the smallest common extension of this sequence and very often it is not.
@article{118716,
author = {Lev Bukovsk\'y and Jaroslav Sk\v riv\'anek},
title = {The smallest common extension of a sequence of models of ZFC},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {35},
year = {1994},
pages = {745-752},
zbl = {0822.03029},
mrnumber = {1321245},
language = {en},
url = {http://dml.mathdoc.fr/item/118716}
}
Bukovský, Lev; Skřivánek, Jaroslav. The smallest common extension of a sequence of models of ZFC. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 745-752. http://gdmltest.u-ga.fr/item/118716/
The model of set generated by countable many generic reals, J. Symbolic Logic 46 (1981), 732-752. (1981) | MR 0641487
Characterization of generic extensions of models of set theory, Fund. Math. 83 (1973), 35-46. (1973) | MR 0332477
A general setting of models extensions, International Workshop on Set theory, Abstracts of the talks, Marseilles-Luminy (1990). (1990)
Generic families and models of set theory with the axiom of choice, Proc. Amer. Math. Soc. 106 (1989), 199-206. (1989) | MR 0994389 | Zbl 0673.03042
Set Theory, Academic Press New York (1978). (1978) | MR 0506523 | Zbl 0419.03028
Iterated Cohen Extensions and Souslin`s Problem, Ann. of Math. 94 (1971), 201-245. (1971) | MR 0294139 | Zbl 0244.02023
Embeddings and automorphisms, Handbook of Boolean algebras J.D. Monk and R. Bonnet North-Holland Amsterdam (1989), 607-635. (1989) | MR 0991604
The Theory of Semisets, Academia Prague (1972). (1972) | MR 0444473