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Topologies on Complete Lattices

Description: One of the more important concepts in mathematics is the concept of order, that is, the description or comparison of two elements of a set in terms of one preceding or being smaller than or equal to the other. If the elements of a set, as pairs, exhibit certain order-type characteristics, the set is said to be a partially ordered set. The purpose of this paper is to investigate a special class of partially ordered sets, called lattices, and to investigate topologies induced on these lattices by specially defined order related properties called order-convergence and star-convergence.
Date: December 1973
Creator: Dwyer, William Karl

A Partial Characterization of Upper Semi-Continuous Decompositions

Description: The goal of this paper is to characterize, at least partially, upper semi-continuous decompositions of topological spaces and the role that upper semi-continuity plays in preserving certain topological properties under decomposition mappings. Attention is also given to establishing what role upper semi-continuity plays in determining conditions under which decomposition spaces possess certain properties. A number of results for non-upper semi-continuous decompositions are included to help clarify the scope of the part upper semi-continuity plays in determining relationships between topological spaces and their decomposition spaces.
Date: December 1973
Creator: Dennis, William Albert

Simplicial Homology

Description: The purpose of this thesis is to construct the homology groups of a complex over an R-module. The thesis begins with hyperplanes in Euclidean n-space. Simplexes and complexes are defined, and orientations are given to each simplex of a complex. The chains of a complex are defined, and each chain is assigned a boundary. The function which assigns to each chain a boundary defines the set of r-dimensional cycles and the set of r—dimensional bounding cycles. The quotient of those two submodules is the r-dimensional homology group.
Date: August 1973
Creator: Chang, Chih-Chen

Some Properties of Ideals in a Commutative Ring

Description: This thesis exhibits a collection of proofs of theorems on ideals in a commutative ring with and without a unity. Theorems treated involve properties of ideals under certain operations (sum, product, quotient, intersection, and union); properties of homomorphic mappings of ideals; contraction and extension theorems concerning ideals and quotient rings of domains with respect to multiplicative systems; properties of maximal, minimal, prime, semi-prime, and primary ideals; properties of radicals of ideals with relations to quotient rings, semi-prime, and primary ideals.
Date: August 1973
Creator: Hicks, Gary B.