Decomposition of mappings between complete lattices by mathematical morphology, Part I. General lattices

Abstract

Two canonical decompositions of mappings between complete lattices are presented. These decompositions are based on the mathematical morphology elementary mappings: erosions, anti-erosions, dilations and anti-dilations. The proposed decompositions are obtained by introducing the concept of morphological connection, that extends the notion of Galois connection. The definitions of sup-generating mapping, kernel and basis within the framework of complete lattices are given. The decompositions are built by analysing the kernel and may be simplified from the basis. The results are specialized to the cases of inf-separable, increasing and decreasing mappings. The presented decompositions are dual. Some examples, including the case of boolean functions simplification, illustrate the key concepts and the decomposition rule.