Eastern Model-Theory for Boolean-Valued Theories
In this paper we study some model-theoretic properties of Boolean-valued models in which the interpretations of certain first-order sentences have prescribed values in a fixed complete Boolean algebra ; these are models of a natural Boolean-valued analogue of a consistent first-order theory as introduced in A. SHORB’S thesis [12] (see also [7]). We obtain conditions for such a Boolean-valued theory to be preserved under submodels or under unions of chains; the conditions are in terms of the least Boolean-value which an universal, or Qhentence, respectively, has to take in a model of the given theory. These results enable us to investigate model completeness and existentially closed models for a Boolean-valued theory. It is proved that the relation between a model companion for a first-order theory and its cIass of existentially closed models still holds in this context.