Year of Award

2008

Document Type

Thesis

Degree Type

Master of Arts (MA)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Chair

George McRae

Commitee Members

Adam Nyman, Joel Henry

Keywords

Conceptual Graph, edge object, loop object, strict morphism, reflective subcategory, vertex object

Publisher

University of Montana

Abstract

In the study of the Category of Graphs, the usual notion of a graph is that of a simple graph with at most one loop on any vertex, and the usual notion of a graph homomorphism is a mapping of graphs that sends vertices to vertices, edges to edges, and preserves incidence of the mapped vertices and edges. A more general view is to create a category of graphs that allows graphs to have multiple edges between two vertices and multiple loops at a vertex, coupled with a more general graph homomorphism that allows edges to be mapped to vertices as long as that map still preserves incidence. This more general category of graphs is named the Category of Conceptual Graphs. We investigate topos and topos-like properties of two subcategories of the Category of Conceptual Graphs. The first subcategory is the Category of Simple Loopless Graphs with Strict Morphisms in which the graphs are simple and loopless and the incidence preserving morphisms are restricted to sending edges to edges, and the second subcategory is the Category of Simple Graphs with Strict Morphisms where at most one loop is allowed on a vertex. We also define graph objects that are their graph equivalents when viewed in any of the graph categories, and mimic their graph equivalents when they are in other categories. We conclude by investigating the possible reflective and corefective aspects of our two subcategories of graphs.

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© Copyright 2008 Demitri Joel Plessas