De Jesus Pagan, Francisco J.

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    Algunas factorizaciones ordenadas
    (2023-05-10) De Jesus Pagan, Francisco J.; Ortiz Albino, Reyes M.; College of Arts and Sciences - Sciences; Castellini, Gabriele; Ocasio Gonzalez, Victor A.; Department of Mathematics; Cabrera Rios, Mauricio
    In 2006, Anderson and Frazier defined the concept of a τ-factorization on an integral domain D. This notion generalized the theory of non-atomic factorizations over integral domains. Anderson and Frazier considered a symmetry relation τ over a set of nonzero and nonunit elements of an integral domain, denoted by D^# and defined a τ-product of two or more elements in D^# to be the products of elements that are related with respect to the relation τ. This notion summarizes many types of factorizations already studied, such as factorizations into primes, co-maximal factorizations, etc. In such theory, they mention that it can be extended to non symmetric relations, such as a partial order relation, but they never developed the idea in such a setting. For a,b∈D^# we define aτ_⊆ b, if (a)⊆(b). Then we can define and study the notion of τ_⊆-factorizations in an analogous way as Anderson and Frazier did for symmetric relations. We study such type of factorizations and some properties as: the existence of the greatest common τ_⊆-factor of any two nonzero nonunit elements, the least common τ_⊆-product, τ_⊆-irreducible elements, among others. We also consider some orders determined by τ_⊆, such as τ_⊂, its dual order τ_⊇ and a covering order. Also, we present some facts about the τ_⊆-irreducible τ_⊆-divisor graph of a nonzero nonunit element. We will focus on a Unique Factorization Domain D. But the reader can use some strategy to extend this to a Non-Unique Factorization Domain.