In this paper, we give a possible characterization of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show that the distributivity is first-order definable on bi-approximation semantics. In addition, we investigate the dual representation of those structures and compare them with bi-approximation semantics for intuitionistic logic. We also discuss that two different methods to validate the distributivity—by the splitters and by the adjointness—can be explicated with the help of the axiom of choice as well.
"The Distributivity on Bi-Approximation Semantics." Notre Dame J. Formal Logic 57 (3) 411 - 430, 2016. https://doi.org/10.1215/00294527-3542442