*3.10. Mathematical Morphology in Stratified Institutions*

The mathematical morphology of [91,92] uses a pair of dual mappings between lattices called 'dilation' and 'erosion' in the context of some mathematical foundations for image analysis. In [53], the authors employ these concepts from mathematical morphology in order to derive pairs of dual connectives. This uses, for a given model *M*, the lattice on the quotient *Sen*(Σ)/≡*<sup>M</sup>* , where *ρ* ≡*<sup>M</sup> ρ* when [[*M*, *ρ*]] = [[*M*, *ρ* ]] and the order on *Sen*(Σ)/≡*<sup>M</sup>* is given by *ρ*/≡*<sup>M</sup>* ≤ *ρ* /≡*<sup>M</sup>* when [[*M*, *ρ*]] ⊆ [[*M*, *ρ* ]]. When the respective stratified institution has conjunctions and disjunctions, (*Sen*(Σ)/≡*<sup>M</sup>* , ≤) is a lattice indeed. The authors provide a general abstract definition of 'dilation' and 'erosion' operators on sentences, *DBρ* and *EBρ*, respectively, which are then extended as operations on *Sen*(Σ)/≡*<sup>M</sup>* . Instances of *DB* and *EB* include the universal and existential quantifications in OFOL as well as the necessity and possibility in various modal logics. Moreover, the authors of [53] develop a general proof theory in stratified institutions based on abstract erosion and dilation operators, which is shown to be complete. Finally, ref. [53] offers some preliminary ideas regarding applications of this theory to qualitative spatial reasoning.
