Date of Award

1999

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

James J. Madden

Abstract

In this work, we explore properties of totally ordered commutative monoids---we call them tomonoids. We build on the work in [E]. Our goal is to obtain results that will be useful for studying totally ordered rings with nilpotents. Chapter 1 presents background information. In Chapter 2, we present some criteria for determining when a tomonoid is a quotient of a totally ordered free monoid by a convex congruence. In Chapter 3, we show that every positive tomonoid of rank 2 is a convex Rees quotient of a subtomonoid of a totally ordered abelian group. In Chapter 4, we provide a classification of all positive nil tomonoids with 6, 7, and 8 elements. From the classification, we deduce that (1) every positive nil tomonoid with 8 or fewer elements is a quotient of a totally ordered free monoid by a convex congruence (2) any positive nil tomonoid with 8 or fewer elements that satisfies a certain weak cancellation law is a convex Rees quotient of a subtomonoid of a totally ordered abelian group. Finally, in Chapter 5 we note some of the known results relating tomonoids and totally ordered rings, and we suggest further research questions.

ISBN

9780599474789

Pages

102

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