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Relación amiga, grafos y generalización
Jiménez Franco, Julián Andrés
Jiménez Franco, Julián Andrés
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Abstract
Anderson y Frazier en 2006 definieron el concepto de factorizaciones generalizadas sobre dominios enteros. Para esto, los autores restringieron la operación multiplicativa de manera que solo permiten multiplicar los elementos que estén relacionados con respecto a una relación simétrica Ƭ. Esto abre puertas a estudiar la teoría de factorizaciones sobre dominios y temas similares desde muchos puntos de vista. Este trabajo considera otra relación de equivalencia sobre Z^#=Z-{0,1,-1} llamada relación amiga.
La relación amiga fue definida en la XXIII Olimpiada Colombiana de Matemática y se denota por R_2= { (x,y) ∈ Z^# | √xy ∈ Z}. Es decir, (x,y) ∈ R_2, si y solo si √xy ∈ Z. Se estudian algunos conceptos de Anderson y Frazier aplicados a la relaci\'on amiga. Entre ellos Ƭ-factorización, elemento Ƭ-irreducible, Ƭ-factorizaciones en Ƭ-irreducibles y el grafo de Ƭ-factores Ƭ-irreducibles. Además, se presenta la relación R_{m/n} como una generalización de R_2.
Anderson and Frazier in 2006 defined the concept of generalized factorizations on integral domains. For this, the authors restricted the multiplicative operation so that they only allow multiplying the elements that are related to a symmetric relation. This opens doors to re-study the theory of factorizations on domains and similar topics from many points of view. This paper considers another equivalence relation over Z<sup>#</sup> = Z−{0,±1} called friend relationship. The friendly relationship was defined at the XXIII Colombian Mathematical Olympiad and is denoted by R<sub>2</sub> = {(x, y) 2 Z<sup>#</sup> Å~ Z<sup>#</sup> | p xy 2 Z}. That is, (x, y) 2 R2, if and only if p xy 2 Z. Some concepts of Anderson and Frazier applied to the friendly relationship are studied. Among them -factorization, element -irreducible, -factorizations in -irreducible and the graph of -factors -irreducible. In addition, the relationship Rm n is presented as a generalization of R<sub>2</sub>.
Anderson and Frazier in 2006 defined the concept of generalized factorizations on integral domains. For this, the authors restricted the multiplicative operation so that they only allow multiplying the elements that are related to a symmetric relation. This opens doors to re-study the theory of factorizations on domains and similar topics from many points of view. This paper considers another equivalence relation over Z<sup>#</sup> = Z−{0,±1} called friend relationship. The friendly relationship was defined at the XXIII Colombian Mathematical Olympiad and is denoted by R<sub>2</sub> = {(x, y) 2 Z<sup>#</sup> Å~ Z<sup>#</sup> | p xy 2 Z}. That is, (x, y) 2 R2, if and only if p xy 2 Z. Some concepts of Anderson and Frazier applied to the friendly relationship are studied. Among them -factorization, element -irreducible, -factorizations in -irreducible and the graph of -factors -irreducible. In addition, the relationship Rm n is presented as a generalization of R<sub>2</sub>.
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2022-07-07
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Relación amiga, Grafos, Factorizaciones generalizadas