Publication:
La composicion de relaciones y la teoria de t-factorizacion
La composicion de relaciones y la teoria de t-factorizacion
Authors
Méndez-Oyuela, David F.
Embargoed Until
Advisor
Ortiz-Albino, Reyes M.
College
College of Arts and Sciences – Sciences
Department
Department of Mathematics
Degree Level
M.S.
Publisher
Date
2018
Abstract
La teoría de t -factorizaciones en dominios integrales fué desarrollada por Anderson
y Frazier [2] en el 2006, la misma caracterizó las factorizaciones conocidas
y abrió las puertas para crear otras. Se puede visualizar como una restricción a la
operación de multiplicación de la estructura; considerando una relación simétrica t
sobre los elementos no invertibles y distintos de cero de un dominio integral. Antes
de formalizar la definición, denotemos D un dominio integral, U(D) el conjunto
de elementos invertibles o unidades de D y D# el conjunto de elementos distintos
de cero que no son unidades de D. Un producto a = a1a2...an es llamado una
t -factorización de a D#, si se cumple que aitaj para todo i ≠j y 2 U(D). A
los elementos ai se les llama t-factores de a y a es llamado un t -producto de los
ai. Note que si t = D# * D#, las t -factorizaciones y las factorizaciones usuales en
D coinciden. Otro ejemplo de relevancia es cuando t = S * S, donde S D# es
un conjunto de elementos distinguidos de D#. De esta manera la teoría generalizó
las factorizaciones en dominios integrales conocidas y estudiadas en años anteriores.
Por ejemplo de las factorizaciones en elementos irreducibles surgieron los dominios
atómicos y de las factorizaciones en elementos primales surgieron los dominios de
Schreier.
Este trabajo tiene como objetivo principal estudiar e investigar el concepto de
t -factorizaciones cuando t es la composición de dos o más relaciones. Este estudio
se puede lograr de dos formas. En la primera se consideran dos relaciones t1, t2 y
se analiza que resultados se pueden obtener sobre la relación t1 o t2. La segunda
forma se basa en tratar de factorizar una relación. Este documento se enfocó más en
la primera forma, detalla algunos elementos de su complejidad, además de observar
como se comportan sus factores, mediante muchos ejemplos.
Para poder trabajar con este concepto, se verifica qué propiedades en especí-
fico se pueden obtener a partir de las relaciones dadas. Entre éstas propiedades se
estudió las más conocidas, reflexividad, simetría, transitividad, antisimetría; y otras
asociadas a la teoría de t-factorizaciones como las relaciones divisivas, que preservan
asociados y multiplicativas. Se presenta una nueva definición de t -factorizaciones y
se demuestran algunos resultados con esta nueva definición.
The theory of t -factorizations on integral domains was developed by Anderson and Frazier [2]. This theory characterized all the known factorizations and opened the opportunity to create new ones. It can be visualized as a restriction to the structure's multiplicative operation, by considering a symmetric relation t on the set of non-zero non-unit elements of an integral domain. Before formalizing the definition, let us denote D to be an integral domain, U(D) the set of units of D and D# the set of non-zero non-units elements of D. The product a = a1a2 an is called a t -factorization of a 2 D#, if aiaj for all i≠j and 2 U(D): The elements ai are called t -factors of a and a is called a t -product of the ai's. If t = D#*D#, the t -factorizations and the factorizations on D coincide. Another example of relevance is when t = S*S, where S is a set of distinguished elements in D#. This is the way the theory generalized all the known factorizations on integral domains. For example, the factorizations into irreducible elements gave the notion of atomic domain and the factorizations into primal elements gave the notion of Schreier domains. The main goal of this work is to study the t -factorization concept, when t is a composition of two or more relations. This study can be achieved in two ways. The first one, is to consider two relations t1, t2 and analyze the results obtained with respect to the relation t1 o t2. The second method is to try to factor a relation. This work focuses more on the first method and shows some details of its complexity with a lot of examples. To achieve this, the specific properties one can obtain from the given relations are verified and analyzed. Some of the studied properties which are the most known include: reflexivity, symmetry, transitivity, antisymmetry. And others related to the t -factorization theory, like: divisive, associate-preserving and multiplicative relations. A new definition for t -factorizations is presented and some results are proven with it.
The theory of t -factorizations on integral domains was developed by Anderson and Frazier [2]. This theory characterized all the known factorizations and opened the opportunity to create new ones. It can be visualized as a restriction to the structure's multiplicative operation, by considering a symmetric relation t on the set of non-zero non-unit elements of an integral domain. Before formalizing the definition, let us denote D to be an integral domain, U(D) the set of units of D and D# the set of non-zero non-units elements of D. The product a = a1a2 an is called a t -factorization of a 2 D#, if aiaj for all i≠j and 2 U(D): The elements ai are called t -factors of a and a is called a t -product of the ai's. If t = D#*D#, the t -factorizations and the factorizations on D coincide. Another example of relevance is when t = S*S, where S is a set of distinguished elements in D#. This is the way the theory generalized all the known factorizations on integral domains. For example, the factorizations into irreducible elements gave the notion of atomic domain and the factorizations into primal elements gave the notion of Schreier domains. The main goal of this work is to study the t -factorization concept, when t is a composition of two or more relations. This study can be achieved in two ways. The first one, is to consider two relations t1, t2 and analyze the results obtained with respect to the relation t1 o t2. The second method is to try to factor a relation. This work focuses more on the first method and shows some details of its complexity with a lot of examples. To achieve this, the specific properties one can obtain from the given relations are verified and analyzed. Some of the studied properties which are the most known include: reflexivity, symmetry, transitivity, antisymmetry. And others related to the t -factorization theory, like: divisive, associate-preserving and multiplicative relations. A new definition for t -factorizations is presented and some results are proven with it.
Keywords
Teoría de t -factorizaciones
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Cite
Méndez-Oyuela, D. F. (2018). La composicion de relaciones y la teoria de t-factorizacion [Thesis]. Retrieved from https://hdl.handle.net/20.500.11801/1706