Electronic Theses and Dissertations

Date of Award

2011

Document Type

Dissertation

Degree Name

M.S. in Mathematics

First Advisor

Laura Sheppardson

Second Advisor

William Staton

Third Advisor

Sandra Spiroff

Abstract

A graph is associated to any commutative ring R where the vertices are the non-zero zero divisors of R with two vertices adjacent if x · y = 0. The zero-divisor graph has also been studied for various algebraic stuctures such as semigroups and partially ordered sets. In this paper, we will discuss some known results on zero-divisor graphs of posets as well as the concept of compactness as it relates to zero-divisor graphs. We will dicuss equivalence class graphs defined on the elements of various algebraic structures and also the reduced graph defined on the vertices of a compact graph. After introducing and discussing some known results on poset dimension, we will show that poset decomposition can be directly related to the equivalence classes represented in a reduced graph. Using this decomposition, we can build a poset of a compact graph with any dimension in a specified interval. Thus we have a device which gives us the ability to study the dimension of a poset of a zero-divisor graph.

Included in

Mathematics Commons

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