Decomposition theory for nonsemimodular lattices

Abstract

in compactly generated atomic lattices was developed which for the most part generalized the classical theory for finite dimensional lattices. It was shown that if a compactly generated atomic lattice is semimodular, then every element has an irredundant meet decomposition into completely irreducible elements. Every element of a compactly generated atomic lattice has a unique irredundant decomposition if and only if the lattice is locally distributive. Aside from uniqueness the most fundamental arithmetical property of irredundant decompositions is the Kurosh-Ore replacement property, that is, for every pair of irredundant decompositions a n Q = nlQ' of an element a and for each irreducible qC-Q there is an irreducible q'EGQ' such that a=q= nn(Q-q). Every modular lattice has this property. Moreover, it was shown that a semimodular, compactly generated, atomic lattice has the replacement property if and only if the lattice is locally modular. All of these results used heavily the semimodular condition. Only the characterization of unique decompositions was carried out for general lattices, and this was accomplished because semimodularity could be proved in this case. The present paper extends the decomposition theory to include nonsemimodular lattices. It is shown in the second section that the elements of any compactly generated atomic lattice have irredundant decompositions. In the third section a necessary and sufficient condition is obtained for an arbitrary compactly generated atomic lattice to have the replacement property. This condition is a modification of the lower-semimodular law, and in the presence of semimodularity is easily seen to be equivalent to local modularity. Throughout this paper the notation and terminology of [I ] is used. Lattice join, meet, inclusion, proper inclusion, and covering, are denoted respectively by the symbols U, C, _, <, and <. The corresponding set operations are