Publication Date
4-23-1973
Abstract
This study is concerned with complete lattices in which order convergence coincides with topological convergence with respect to the order topology. Such lattices are termed convergence lattices. After a brief discussion of topology on lattices (Introduction), some preliminary results concerning order convergence and the order topology are presented (Section 1). Several theorems characterizing convergence lattices are presented in Section 2. There it is shown that every neighborhood of a point of a convergence lattice contains an interval that is also a neighborhood of the point. It is also shown that each complete chain, each arbitrary product of convergence lattices, and each lattice in which each chain is finite is a convergence lattice. Various examples of important convergence lattices and some complete lattices failing to be convergence lattices are given in Section 3. Section 4 discusses compactness in convergence lattices. Theorem: A convergence lattice is compact if and only if the interval topology is Hausdorff. Theorem: Every completely distributive complete lattice is a compact convergence lattice. Theorem: Every locally compact convergence lattice is locally convex. Theorem: A complete Boolean algebra is a convergence lattice if and only if the order and interval topologies coincide. (This last theorem completely solves Birkhoff's Problem 76 [1].) Section 5 collects six unsolved problems.
Degree Name
Mathematics
Level of Degree
Doctoral
Department Name
Mathematics & Statistics
First Committee Member (Chair)
Ralph Elgin DeMarr
Second Committee Member
Arthur Steger
Third Committee Member
Liang-Shin Hahn
Language
English
Document Type
Dissertation
Recommended Citation
Gingras, Armando Rosario. "Convergence Lattices." (1973). https://digitalrepository.unm.edu/math_etds/253