On ultrafilter extensions of first-order models and ultrafilter interpretations

There exist two known types of ultrafilter extensions first-order models, both in a certain sense canonical. One them (Goranko Filter and structures: universal-algebraic aspects, preprint, 2007) comes from modal logic universal algebra, fact goes back to Jónsson Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) model theory algebra ultrafilters, with semigroups (Hindman Strauss Algebra the Stone–Čech Compactification, W. de Gruyter, Berlin, as its main precursor. By classical general topology, space ultrafilters over discrete is largest compactification. result (Lect 2012), which confirms canonicity this extension, generalizes spaces endowed an arbitrary structure. An analogous for former type was obtained (in On binary relations. arXiv:2001.02456 ). Results such kind are referred extension theorems. After brief introduction, we offer uniform approach based on idea extend procedure itself. We propose generalization standard concept interpretations functional relational symbols interpreted rather by sets functions relations than themselves, define models appropriate semantics them. provide specific operations turn into ordinary establish necessary sufficient conditions under latter canonical some obtain topological characterization models. generalize restricted version theorem To formulate full version, wider their limits show that can be identified, way, particular case latter; moreover, new absorbs more narrow ones wide sense, images these operations, also Finally, three versions sense. results first sections paper were partially announced Poliakov (in: Kennedy, Queiroz concepts generalizations, Springer, 2017).