Boolean universes above Boolean models

We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory, and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are ``boundedly algebraically compact" in the language $( + , - , . , \wedge , \vee , \leq )$, and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language $( + , \vee , \leq )$. We also give an example with any first-order language. The proofs can be translated into ``naive set theory" in a uniform way