Abstract

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 $$, and the positive cone of a complete $l$-group with infinity adjoined is algebraically compact in the language $$. We also give an example with any first-order language. The proofs can be translated into "naive set theory" in a uniform way.