  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.
