Connectivity on Complete Lattices

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. A non topological axiomatics has been proposed by Matheron and Serra. The present paper extends it to complete sup-generated lattices. A connection turns out to be characterized by a family of openings labelled by the sup-generators, which partition each element of the lattice into maximal terms, of zero infima. When combined with partition closings, these openings generate strong sequential alternating filters. Starting from a first connection several others may be designed by acting on some dilations or symmetrical operators.When applying this theory to function lattices, one interprets the so-called connected operators in terms of actual connections, as well as the watershed mappings. But the theory encompasses the numerical functions and extends, among others, to multivariate lattices. Keywords : connectivity, complete lattices, mathematical morphology, connected operators, filters b...

• Type: Journal
• Year: 1998
• Where: JMIV
• Authors: Jean Serra
• Keywords: JMIV 1998, Non Topological Axiomatics, Sup-generated Lattices, Topological Notion