## Publication: La composicion de relaciones y la teoria de t-factorizacion

##### Authors
Méndez-Oyuela, David F.
Ortiz-Albino, Reyes M.
##### College
College of Arts and Sciences – Sciences
##### Department
Department of Mathematics
M.S.
2018
##### Abstract

The theory of t -factorizations on integral domains was developed by Anderson and Frazier [2]. This theory characterized all the known factorizations and opened the opportunity to create new ones. It can be visualized as a restriction to the structure's multiplicative operation, by considering a symmetric relation t on the set of non-zero non-unit elements of an integral domain. Before formalizing the definition, let us denote D to be an integral domain, U(D) the set of units of D and D# the set of non-zero non-units elements of D. The product a = a1a2 an is called a t -factorization of a 2 D#, if aiaj for all i≠j and 2 U(D): The elements ai are called t -factors of a and a is called a t -product of the ai's. If t = D#*D#, the t -factorizations and the factorizations on D coincide. Another example of relevance is when t = S*S, where S is a set of distinguished elements in D#. This is the way the theory generalized all the known factorizations on integral domains. For example, the factorizations into irreducible elements gave the notion of atomic domain and the factorizations into primal elements gave the notion of Schreier domains. The main goal of this work is to study the t -factorization concept, when t is a composition of two or more relations. This study can be achieved in two ways. The first one, is to consider two relations t1, t2 and analyze the results obtained with respect to the relation t1 o t2. The second method is to try to factor a relation. This work focuses more on the first method and shows some details of its complexity with a lot of examples. To achieve this, the specific properties one can obtain from the given relations are verified and analyzed. Some of the studied properties which are the most known include: reflexivity, symmetry, transitivity, antisymmetry. And others related to the t -factorization theory, like: divisive, associate-preserving and multiplicative relations. A new definition for t -factorizations is presented and some results are proven with it.
##### Keywords
Teoría de t -factorizaciones
##### Persistent URL
Méndez-Oyuela, D. F. (2018). La composicion de relaciones y la teoria de t-factorizacion [Thesis]. Retrieved from https://hdl.handle.net/20.500.11801/1706