Publication:
Factorizaciones donde cada factor de un elemento pertenence a solo una clase de equivalencia
Factorizaciones donde cada factor de un elemento pertenence a solo una clase de equivalencia
Authors
Serna-Rapello, Cesar
Embargoed Until
Advisor
Ortiz-Albino, Reyes M.
College
College of Arts and Sciences - Sciences
Department
Department of Mathematics
Degree Level
M.S.
Publisher
Date
2014-08
Abstract
En el año 2006, Anderson y Frazier desarrollaron la teoría de τ -factorizaciones sobre dominios integrales, como una extensión de las factorizaciones comaximales de McAdam y Swan. La teoría de τ -factorizaciones generaliza el concepto de factorizaciones en la forma usual, y consiste en restringir el dominio de la operación producto de un dominio integral D a una relación simétrica sobre el conjunto de elementos distintos de cero y no unidades de D. Esta teoría ha sido estudiada por Hamon, Ortiz-Albino, Juett, Florescu, entre otros. Este trabajo se enfoca en estudiar la teoría de τ -factorizaciones cuando τ es una relación de equivalencia. Existen diversas razones para considerar relaciones de equivalencia en el estudio de las τ -factorizaciones, la principal es el historial de importacia no sólo en el álgebra abstracta, si no también en las matemáticas en general. Por el aspecto de la teoría, considerar relaciones de equivalencia marca el estudio de relaciones que no son ”artificiales”, es decir, aquellas relaciones cuyo dominio es casi toda la estructura algebraica (esto es más difícil que lo que los autores principales consideraron). Además, los principales resultados en la teoría de τ -factorizaciones, se han obtenido para relaciones divisibles; cuando se trabaja con relaciones de equivalencia no se puede asumir la propiedad de divisibilidad de la relación. Por último, hallar ejemplos de relaciones de equivalencia multiplicativas y que preserven asociados. Hamon estudió parcialmente las relaciones de equivalencia módulo n en el dominio de los números enteros. En base a su investigación y otros acercamientos de Juett, se creía que no había muchos ejemplos de relaciones de equivalencia que sean multiplicativas y preserven asociados. En este trabajo se presentan tres familias infinitas de relaciones de equivalencia que satisfacen estas condiciones. Además, se proveen resultados de propiedades de factorizaciones, bajo estas condiciones, demostrando que ser una relación divisible es una condición suficiente, pero no necesaria para hacer τ -refinamientos. Además, se estudian relaciones de equivalencia que no necesariamente satisfacen la propiedad de preservar asociados. Cuando se extendieron las relaciones de equivalencia unitarias a relaciones de equivalencia que preservan asociados, se obtuvo que las propiedades de factorizaciones, tanto en la relación original como en la de su extensión, son equivalentes. En conclusión, se puede asumir que si la relación de equivalencia es unitaria, preserva asociados. Al factorizar, se obtiene un resultado equivalente al que posee la relación unitaria que no preserva asociados. Por último, se presenta una extensión multiplicativa para las relaciones de equivalencia módulo n. La misma puede servir de modelo para hallar varias propiedades y se presentan algunos posibles trabajos futuros con respecto a este tipo de extensiones de relaciones de equivalencia.
In 2006, Anderson and Frazier developed the theory of τ -factorizations on integral domains, as a extension of the McAdam and Swan’s comaximal factorizations. This theory is a generalization of the usual theory of factorization, which is based on restricting the domain of the usual multiplicative operation of the integral domain to a symmetric relation on the nonzero nonunit elements of D. This theory has been studied by Hamon, Ortiz, Juett, Florescu, and others. This work’s main focus is to study the theory of τ -factorizations, when τ is an equivalence relation. There are reasons why equivalence relations into study of τ -factorizacions, the main reason is the history and importance of equivalence relations in abstract algebra and in mathematics in general. From the point view of this theory, the study of τ -factorization with relations that are not “ artificial”, that is relations whose domain is almost the algebraic structure (this setting is more difficult than the work done by previous authors). Moreover, the main results in the theory of τ -factorizations were obtained assuming the relation was divisible; such hypothesis can not be assume when working with equivalence relations. Hamon studied some topics of the equivalence relation τ(n) , that is, the relation modulo n on the set of integers. Based on this work and other studies by Juett, it was thought that there were very few examples of multiplicative equivalence relations that preserve associates. In this report, three infinite families of such types of relations are shown. Some results of factorization properties under these types of relations are given. Moreover, such results give evidence to show that divisible relations is a sufficient, but not a necessary condition to obtain τ -refinements. Additionally, the equivalence relations that do not satisfy the associated-preserving condition were studied. The associated-preserving extension of a unitary equivalence relation, which does not preserves associates, has exactly the same equivalent factorization properties as the original relation. In conclusion, there is no harm in assuming that a unitary equivalence relation may preserve associates. Finally, a multiplicative extension of the equivalence relation modulo n is presented. This work is an example of the begining of the study of this type of extension. Hence some properties and observations regarding this work could result in future work for multiplicative extensions of an arbitrary equivalence relation.
In 2006, Anderson and Frazier developed the theory of τ -factorizations on integral domains, as a extension of the McAdam and Swan’s comaximal factorizations. This theory is a generalization of the usual theory of factorization, which is based on restricting the domain of the usual multiplicative operation of the integral domain to a symmetric relation on the nonzero nonunit elements of D. This theory has been studied by Hamon, Ortiz, Juett, Florescu, and others. This work’s main focus is to study the theory of τ -factorizations, when τ is an equivalence relation. There are reasons why equivalence relations into study of τ -factorizacions, the main reason is the history and importance of equivalence relations in abstract algebra and in mathematics in general. From the point view of this theory, the study of τ -factorization with relations that are not “ artificial”, that is relations whose domain is almost the algebraic structure (this setting is more difficult than the work done by previous authors). Moreover, the main results in the theory of τ -factorizations were obtained assuming the relation was divisible; such hypothesis can not be assume when working with equivalence relations. Hamon studied some topics of the equivalence relation τ(n) , that is, the relation modulo n on the set of integers. Based on this work and other studies by Juett, it was thought that there were very few examples of multiplicative equivalence relations that preserve associates. In this report, three infinite families of such types of relations are shown. Some results of factorization properties under these types of relations are given. Moreover, such results give evidence to show that divisible relations is a sufficient, but not a necessary condition to obtain τ -refinements. Additionally, the equivalence relations that do not satisfy the associated-preserving condition were studied. The associated-preserving extension of a unitary equivalence relation, which does not preserves associates, has exactly the same equivalent factorization properties as the original relation. In conclusion, there is no harm in assuming that a unitary equivalence relation may preserve associates. Finally, a multiplicative extension of the equivalence relation modulo n is presented. This work is an example of the begining of the study of this type of extension. Hence some properties and observations regarding this work could result in future work for multiplicative extensions of an arbitrary equivalence relation.
Keywords
Factorizations,
Equivalence
Equivalence
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Persistent URL
Cite
Serna-Rapello, C. (2014). Factorizaciones donde cada factor de un elemento pertenence a solo una clase de equivalencia [Thesis]. Retrieved from https://hdl.handle.net/20.500.11801/125