ON THE LATTICE OF SUBLATTICES OF A FINITE DISTRIBUTIVE LATTICE

Abstract

The objective of this thesis is to study the structure of the lattice S(L) of sublattices of a finite distributive lattice L and the interdependence between the lattices Land S(L). This thesis consists of four chapters. Chapter 1 gives the basic terms and notations that are needed throughout the thesis. It includes some important known results which should prove helpful to the study of the chapters that follow. Chapter 2 is concerned with the grading number g(L) which is defined as the absolute value of the difference between the length of a longest maximal chain and that of a shortest maximal chain in S(L). A characterization theorem on finite distributive lattices L satisfying the equation g(L) = 1 is established. In Chapter 3, the grading number of any finite Boolean lattice is determined. This result is then utilized to ascertain the grading numbers of all direct products of finitely many finite chains. In Chapter 4, the concepts of purity and double purity of the lattice are defined in terms of the Frattini sublattice of a lattice. A sufficient condition on a finite distributive lattice L whereby S(L) is pure is provided and the structure of any finite distributive lattice S(L)is doubly pure is worked out. As a conclusion, some related problems on the structure of S(L) are suggested for further study.