Adjunctions on the lattices of partitions and of partial partitions

Abstract

The complete lattice Π(E) of partitions of a space E has been extended into Π*(E), the one of partial partitions of E (where the space covering axiom is removed). We recall the main properties of Π*(E), and exhibit two adjunctions (residuations) between Π(E) and Π*(E). Given two spaces E 1 and E 2 (distinct or equal), we analyse adjunctions between Π*(E 1) and Π*(E 2), in particular those where the lower adjoint applies a set operator to each block of the partial partition; we also show how to build such adjunctions from adjunctions between $${\mathcal{P}(E_2)}$$ and $${\mathcal{P}(E_2)}$$ (the complete lattices of subsets of E 1 and E 2). They are then extended to adjunctions between Π(E 1) and Π(E 2). We obtain as particular case the adjunction on Π(E) that was defined by Serra (for the upper adjoint) and Ronse (for the lower adjoint). We also study dilations from Π*(E 1) to an arbitrary complete lattice L; a particular case is given, for $${L \subseteq [0,+\infty]}$$ , by ultrametrics; then the adjoint erosion provides the corresponding hierarchy. We briefly discuss possible applications in image processing and in data clustering.