Boolean universes above Boolean models

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: Atom; Boolean model; first-order language; convergence in lattice-ordered rings; equational compactness; algebraic compactness

Domaines: Mathématiques générales [math.GM]; Catégories et ensembles [math.CT]; Logique [math.LO]

Friedrich Wehrung. Boolean universes above Boolean models. The Journal of Symbolic Logic, 1993, 58, no. 4, pp.1219-1250. ⟨hal-00004693⟩