## Publication: Imagen de los τ-Productos

 dc.contributor.advisor Ortiz Albino, Reyes M. dc.contributor.author Calderón Gómez, José E. dc.contributor.college College of Arts and Sciences - Art en_US dc.contributor.committee Ocasio, Victor dc.contributor.committee Dziobiak, Stan dc.contributor.department Department of Mathematics en_US dc.contributor.representative Irizarry, Zollianne dc.date.accessioned 2019-07-01T17:47:05Z dc.date.available 2019-07-01T17:47:05Z dc.date.issued 2019-05-15 dc.description.abstract The theory of $\tau$-factorizations, also known as theory of generalized factorizations, was developed by Anderson and Frazier in 2006. It was the result of a generalization of the comaximal factorizations by McAdams and Swam, replacing the condition of being comaximals to being related on the set of nonzero nonunits elements in the integral domain. Denote $D$ as an integral domain, $U(D)$ as the set of units of $D$ and $D^\#$ as the set of elements nonzero nonunits of $D$. The authors considered symmetric relations defined over the nonzero nonunits elements. The usual theory of factorizations came to be a particular case, where the relation used is $\tau=D^\#\times D^\#$. An expression of the form $a=\lambda a_1\cdot\cdot \cdot a_n$, where $\lambda\in U(D)$ and $a_i\tau a_j$ for all $1\leq i\neq j\leq n$, is called a $\tau$-factorizarion of $a$. Each $a_i$ is called a $\tau$-factor of $a$ and $a$ is a $\tau$-product of $a_i$. Furthermore, it is possible to obtain particular cases, such as factorizations in irreducibles elements, primals, and others, by taking $\tau=S\times S$, where $S$ is the set of irreducible elements or primals respectively. This work studied the relation $\tau_R$, where $R\subseteq D\times E$, $D$ and $E$ are integral domains, and $\tau$ is defined on $D^\#$. The relation $\tau_R$ is defined as $x\tau_R y$, if and only if there exist $a,b\in D^\#$ such that $a\tau b$, $aRx$, and $bRy$. That is, $\tau_R$ is the image of $\tau$ with respect to the relation $R$''. The properties of $\tau_R$ that can be inherited from $\tau$ in $\tau_R$ are analyzed . It must be clarified that although the definition is given with respect to the image of a relation, most of the work is focused in different types of functions, such as one to one and surjectives functions, homomorphisms, and others. The principal objective is to provide a way to study $\tau$-factorizations and structural properties using the images of the functions. en_US dc.description.graduationSemester Spring en_US dc.description.graduationYear 2019 en_US dc.identifier.uri https://hdl.handle.net/20.500.11801/2482 dc.language.iso es en_US dc.rights Attribution-NonCommercial-NoDerivatives 4.0 International * dc.rights.holder (c) 2019 José Emilio Calderón en_US dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ * dc.subject factorizaciones en_US dc.subject Relaciones en_US dc.subject Dominios de integridad en_US dc.subject Función en_US dc.subject Homomorfismo en_US dc.subject.lcsh Factorization (Mathematics) en_US dc.subject.lcsh Integral domains en_US dc.subject.lcsh Functions en_US dc.title Imagen de los τ-Productos en_US dc.type Thesis en_US dspace.entity.type Publication thesis.degree.discipline Pure Mathematics en_US thesis.degree.level M.S. en_US
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