BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Krzysztof Mierzewski (Carnegie Mellon University) DTSTART:20250508T180000Z DTEND:20250508T190000Z DTSTAMP:20260423T021440Z UID:OLS/179 DESCRIPTION:Title: Th e logics of kernels and closures\nby Krzysztof Mierzewski (Carnegie Me llon University) as part of Online logic seminar\n\n\nAbstract\nEach subse t D of a complete Boolean algebra generates both a closure operator and a kernel operator on the algebra\, respectively mapping each element to its lower approximation (the join of all D-elements below it) and its upper ap proximation (the meet of all D-elements above it). I will discuss the bimo dal logics of such approximation operators on Boolean algebras. By varying the constraints imposed on the generating set D\, we obtain a natural fam ily of modal logics. The resulting algebraic approximation semantics offer s a new perspective on several common modal systems: I will show how well- known modal logics can be recovered as logics of approximation for particu lar choices of constraints on the generating set\, and one can trace the e mergence of various modal laws to simple structural features of the genera ting set. The logic of approximation operators generated by arbitrary subs ets D is the subnormal logic EMNT4+EMNT4. I will give a simple criterion t hat characterizes the corresponding class of algebras: that is\, algebras with abstract closure and kernel operators that are representable as appro ximation operators. The complete logic of approximation operators generate d by a sublattice is the fusion S4+S4: the completeness result relies on a correspondence between sublattice-generated approximation operators and p airwise zero-dimensional bitopological spaces. When D is a complete sublat tice\, we obtain exactly the temporal logic S4t. When D is a subalgebra\, the two modalities collapse into one as they become each other’s duals\, and we obtain monomodal S5 (for complete subalgebras) and S4 (for subalge bras in general).\n LOCATION:https://researchseminars.org/talk/OLS/179/ END:VEVENT END:VCALENDAR