Many-sorted coalgebraic modal logic: A model-theoretic study (Q2747941)

A semantical study of many-sorted modal logic associated with certain dynamical systems is given, which applies the approach of \textit{B. Jacobs} [``Towards a duality result in coalgebraic modal logic'', Electronic Notes in Theoretical Computer Science 33 (2000; Zbl 0959.03503)] to the many-sorted modal logic of \textit{M. Rößiger} [``Coalgebras and modal logic'', ibid. (2000; Zbl 0959.03502)]. Sort-indexed formulas of many-sorted modal logic are interpreted as predicates in a coalgebra, and also as elements of Boolean algebras. The main semantical structures of the paper are the many-sorted Boolean algebras with operators (MBAO) which are indexed by sorts, following general ideas from categorical logic; after the interpretation of the logic in the coalgebras is given, the completeness follows from a Lindenbaum construction. Later, reverse translations from MBAO to coalgebras are presented, one of them as an algebraic reformulation of Rößiger's construction, and another more natural one which gives rise to an ultrafilter extension result and a final coalgebra.