A General Dependence Relation and its Application to Lattice Imbeddings

It is well known that a dependence relation defined between the elements and the subsets of an abstract set M can be used to construct a complete lattice L' The elements of L' are the subsets of M which are closed with respect to the dependence relation. The properties of L' are determined by the set M and the dependence relation. If the set M is taken to be a set of lattice elements, a partial ordering is defined over M by the lattice ordering. In this thesis postulates are given for a generalized dependence relation which takes into account any partial ordering which is defined over M and which reduces to the classical dependence relation if M is not ordered. In particular if M is taken to be the set of join irreducible elements of a lattice L, then the complete lattice L', which is induced by a generalized dependence relation, is such that the set of completely join irreducible elements of L' is isomorphic to M. As the dependence relation is varied, different lattices are obtained, all of which have the same set of join irreducible elements. Let L be any finite dimensional lattice over which an integral valued semi-modular function σ defined. In Part II the theory of Part I is applied to imbed Las a sublattice of a semi-modular lattice L' such that if a → a', then the ordinary lattice rank of a' equals σ(a). In Part III the following imbedding problem is discussed. If a given lattice L has the property that every quotient lattice u/a for a ≠ z in Lis distributive (modular, semi-modular), is it always possible to extend L to a distributive (modular, semi-modular) lattice L' by introducing new elements which contain no element of L except z? It is shown that the process is always possible in the finite dimensional distributive case and that the resulting lattice L' is unique under an additional mild restriction. However, for the modular and semi-modular cases, counter examples are given to prove that in general the imbedding is impossible.