Boolean indexed models and Wheeler's conjecture

Wheeler conjectured that if a theory has a model companion, then its universal Horn fragment has a model companion. This conjecture was based on several examples, such as the universal Horn fragments of the theories of commutative integral domains and ordered commutative integral domains. In models of these universal Horn fragments there is a definable underlying Boolean algebra. Wheeler's Conjecture was proven false by Glass and Pierce. We prove a positive alternative to Wheeler's Conjecture by introducing a translation to a language over which models have an underlying Boolean algebra. This translation mimics the supportive examples. We also discuss the logical strength of this translation.