On two types of ultrafilter extensions of binary relations

Abstract: There exist two distinct types of ultrafilter extensions of binary relations, one discovered in universal algebra and modal logic, and another, in model theory and algebra of ultrafilters. We show that the extension of the latter type is properly included in the extension of the former type, and describe their interaction with the relation algebra operations. Then we provide topological characterizations of both extensions and show that the larger extension continuously maps the space of ultrafilters into the space of filters endowed with the Vietoris topology.