LATTICES WITH UNIQUE IRREDUCIBLE DECOMPOSITIONS

We define the concept of a minimal decomposition in a lattice, and prove that all the currently known lattices with unique irreducible decompositions are in fact lattices with minimal ones. Also. the characterization of a class of lattices with minimal decompositions is given. A new proof of the Crawley-Dilworth characterization theorem for the class of coalgebraic strongly coatomic lattices with unique irreducible decompositions obtains as a consequence.