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Grafos de factores en dominios con integridad
Romero Castro, Offir Neil
Romero Castro, Offir Neil
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Abstract
Anderson y Frazier (2006) definieron la teoría de τ-factorizaciones o de factorizaciones generalizadas utilizando una relación simétrica τ sobre D<sup>#</sup>, el conjunto de elementos distintos de cero y de unidades de un dominio con integridad D. La idea se puede interpretar como el estudio del producto de elementos que se relacionan con respecto a τ. Este concepto generalizó muchos casos de factorizaciones previamente estudiadas como: factorizaciones primas, factorizaciones en elementos irreducibles, factorizaciones comaximales, entre otras. Sea P(D<sup>#</sup>) el conjunto potencia de D<sup>#</sup>, α ∈ P(D<sup>#</sup>) y τ una relación simétrica sobre D<sup>#</sup>. Este trabajo considera la subrelación τ<sup>α</sup> = {(x, y) : x, y ∈ α y xτy} de τ y presenta algunas de sus características y propiedades. Se define el grafo de factores de un elemento en D<sup>#</sup> como una generalización del grafo de divisores irreducibles de Coykendall y Maney, del grafo de τ-factores τ-irreducibles y del grafo de α-β-divisores de Mooney. Se muestran algunos ejemplos y propiedades de los grafos de factores, así como las implicaciones de la relación τ<sup>α</sup> en sus subgrafos.
Anderson and Frazier (2006) defined the theory of τ-factorizations or theory of generalized factorizations, using a symmetric relation τ over D<sup>#</sup>, the set of elements distinct to zero and the units of an integral domain D. This idea can be interpreted as the study of the factorization of elements that are related with respect to a symmetric relation τ. This concept generalized many cases of factorizations previously studied, such as prime factorizations, factorizations in irreducible elements, comaximal factorizations, and more. Let P(D<sup>#</sup>) be the power set of D<sup>#</sup>, α ∈ P(D<sup>#</sup>) and τ a symmetic relation over D<sup>#</sup>. This research considers the subrelation τ<sup>α</sup> = {(x, y) : x, y ∈ α and xτy} of τ and presents some of its characteristics and properties. It is defined the graph of factors of an element of D<sup>#</sup> as a generalization of the graph of irreducible divisors of Coykendall and Maney, the graph of τ-irreducible τ-factors and the graph of α-β-divisors of Mooney. It is shown some examples and properties of the graphs of factors, and the implications of the relation τ<sup>α</sup> in their subgraphs.
Anderson and Frazier (2006) defined the theory of τ-factorizations or theory of generalized factorizations, using a symmetric relation τ over D<sup>#</sup>, the set of elements distinct to zero and the units of an integral domain D. This idea can be interpreted as the study of the factorization of elements that are related with respect to a symmetric relation τ. This concept generalized many cases of factorizations previously studied, such as prime factorizations, factorizations in irreducible elements, comaximal factorizations, and more. Let P(D<sup>#</sup>) be the power set of D<sup>#</sup>, α ∈ P(D<sup>#</sup>) and τ a symmetic relation over D<sup>#</sup>. This research considers the subrelation τ<sup>α</sup> = {(x, y) : x, y ∈ α and xτy} of τ and presents some of its characteristics and properties. It is defined the graph of factors of an element of D<sup>#</sup> as a generalization of the graph of irreducible divisors of Coykendall and Maney, the graph of τ-irreducible τ-factors and the graph of α-β-divisors of Mooney. It is shown some examples and properties of the graphs of factors, and the implications of the relation τ<sup>α</sup> in their subgraphs.
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Date
2022-07-07
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Keywords
Grafos de factores, Factorizaciones generalizadas, Graphs of factors, Generalized factorizations