Decomposition theory for lattices without chain conditions

Abstract

In this thesis we are concerned with representing an element of a lattice as an irredundant meet of elements which are irreducible in the sense that they are not proper meets, and with certain arithmetical properties of these decompositions. A theory is developed for the class of compactly generated atomic lattices which extends the classical theory for finite dimensional lattices. The principal results are the following. Every element of an arbitrary compactly generated atomic lattice has an irredundant meet decomposition into irreducible elements. These decompositions are unique in distributive lattices. In a modular lattice the decompositions of an element have the Kurosh-Ore replacement property, that is, for any two decompositions of an element, each irreducible in the first decomposition can be replaced by a suitable irreducible in the second decomposition. Moreover, characterizations are obtained of those lattices having unique decompositions and those lattices having the replacement property.