A type of generalized factorization on domains -Factorizations R.M. - PowerPoint PPT Presentation

Notation and Definitions Equivalence relations A type of generalized factorization on domains τ -Factorizations R.M. Ortiz-Albino Conference of rings and factorizations, 2018 R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Equivalence relations Outline Notation and Definitions 1 Definitions Relations Equivalence relations 2 Motivation Some results (Ortiz and Serna) R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Notation D denotes an integral domain D ♯ is the set of nonzero nonunits elements of D τ denotes a symmetric relation on D ♯ R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Outline Notation and Definitions 1 Definitions Relations Equivalence relations 2 Motivation Some results (Ortiz and Serna) R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Definition of a τ -Factorization Definition We say x ∈ D ♯ has a τ -factorization if x = λ x 1 ··· x n where λ is a unit in D and x i τ x j for each i � = j . We say x is a τ -product of x i ∈ D ♯ and each x i is a τ -factor of x (we write x i | τ x ). Vacuously, x = x and x = λ · ( λ − 1 x ) are τ -factorizations, known as the trivial ones. Definition In general, x | τ y (read x τ -divides y ) means y has a τ -factorization with x as a τ -factor. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Definition We call x ∈ D ♯ a τ -atom, if the only τ -factorizations of x are of the form λ ( λ − 1 x ) (the trivial τ -factorizations ). Example: Irreducible elements are τ -atoms (for any relation τ on D ♯ ). Definition A τ -factorization λ x 1 ··· x n is a τ -atomic factorization if each x i is a τ -atom. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Definition If you exchange the factorization, irreducible elments and divide operator by τ -factorization, τ -atom and | τ operator in the definitions of GCD domain, UFD, HFD, FFD, BFD and ACCP. We obtain the notions of: τ -GCD domain, τ -UFD, τ -HFD, τ -FFD, τ -BFD, and τ -ACCP. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Examples Example Let τ D = D ♯ × D ♯ (the greatest relation), then the τ D -factorizations are the usual factorizations. Example Let τ / 0 = ∅ (the trivial), then we have that every element is a τ -atom. So any integral domain is in fact a τ / 0 -UFD. Example Let τ S = S × S , where S ⊂ D ♯ . Hence you can consider S is the set of primes, irreducible, primals, primary elements, rigid elements,... R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Outline Notation and Definitions 1 Definitions Relations Equivalence relations 2 Motivation Some results (Ortiz and Serna) R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Types of Relations Definitions and Properties Let x , y , z ∈ D ♯ . Associate We say τ is associate - preserving if preserving x τ y and y ∼ z implies x τ z . Divisive If λ x 1 ··· x n is a τ -fatorization, then Multiplicative x 1 ··· x i − 1 · ( λ x i ) · x i +1 ··· x n is also a τ -factotization. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Types of Relations Definitions and Properties Let x , y , z ∈ D ♯ . We say τ is divisive if x τ y and z | x implies z τ y . Associate Divisive implies associate-preserving. preserving If τ is divisive, then we can do Divisive τ -refinements. Multiplicative That is, if x 1 ··· x n is a τ -factorization and z 1 ··· z m is a τ -factorization of x i then x 1 ··· x i − 1 · z 1 ··· z m · x i +1 ··· x n is also a τ -factorization. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Types of Relations Definitions and Properties Let x , y , z ∈ D ♯ . Associate We say τ is multiplicative if x τ y and preserving x τ z implies x τ ( yz ). Divisive If τ is multiplicative, then each Multiplicative nontrivial τ -factorization can be written into a τ -product of length 2 R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations More Examples Example Let τ ( n ) = { ( a , b ) | a − b ∈ ( n ) } a relation on Z ♯ , for each n ≥ 0. For n = 1, we obtained the usual factorizations. Note that for n ≥ 2, τ ( n ) is never divisive, but it is multiplicative and associate-preserving for n = 2. (Hamon) Z is a τ ( n ) -UFD if and only if n = 0 , 1. (Hamon and Juett) Z is a τ ( n ) -atomic domain if and only if n = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 8 , 10. (Ortiz) Z is a τ ( n ) -GCD domain if and only if is a τ ( n ) -UFD. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations More Examples Example Let ∗ be a start-operation on D . Then define ⇒ ( x , y ) ∗ = D , that is, x and y are ∗ -comaximal. x τ ∗ y ⇐ It is both multiplicative and divisive. If ⋆ = d , then a d -factorization is the known comaximal factorization defined by McAdam and Swam. R. M. Ortiz-Albino τ -Factorization

� � � � � � � � � � � � Notation and Definitions Definitions Equivalence relations Relations Diagram of Properties (Anderson and Frazier) UFD FFD BFD ACCP atomic ∗ τ -FFD ∗ ∗ ∗ ∗ � τ -ACCP � τ -atomic τ -UFD τ -BFD τ -HFD Figure: Diagram of structures and τ -structures, when τ is divisive. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Definitions Equivalence relations Relations Definition of “ ≤ ” Definition We say τ 1 ≤ τ 2 , if τ 1 ⊆ τ 2 as sets. Theorem Let D be an integral domain and τ 1 , τ 2 be two relations on D ♯ . The following are equivalent: τ 1 ≤ τ 2 . For any x , y ∈ D ♯ , x τ 1 y ⇒ x τ 2 y. Any τ 1 -factorization is a τ 2 -factorization. R. M. Ortiz-Albino τ -Factorization

� � � � � � � � � � � � Notation and Definitions Definitions Equivalence relations Relations Theorem (Ortiz) τ 2 -UFD τ 2 -FFD τ 2 -BFD τ 2 -ACCP τ 2 -atomic τ 1 -FFD � τ 1 -ACCP � τ 1 -atomic τ 1 -UFD τ 1 -BFD τ 1 -HFD Figure: Properties when τ 1 ⊆ τ 2 both divisive and τ 2 multiplicative R. M. Ortiz-Albino τ -Factorization

� � � � � � � � � � � � Notation and Definitions Definitions Equivalence relations Relations Theorem(Juett) τ 2 -UFD τ 2 -FFD τ 2 -BFD τ 2 -ACCP τ 2 -atomic τ 1 -FFD � τ 1 -ACCP � τ 1 -atomic τ 1 -UFD τ 1 -BFD τ 1 -HFD Figure: Properties when τ 1 ⊆ τ 2 , τ 1 divisive and τ 2 refinable and associated-preserving R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Motivation Equivalence relations Some results (Ortiz and Serna) Outline Notation and Definitions 1 Definitions Relations Equivalence relations 2 Motivation Some results (Ortiz and Serna) R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Motivation Equivalence relations Some results (Ortiz and Serna) Equivalence relations have historical precedent. Equivalence relation are less artificial relations. There is only one divisive equivalence relation τ D . Divisive seems to be more-less understood to be good type of relation. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Motivation Equivalence relations Some results (Ortiz and Serna) Outline Notation and Definitions 1 Definitions Relations Equivalence relations 2 Motivation Some results (Ortiz and Serna) R. M. Ortiz-Albino τ -Factorization

� � � � � � � � � � Notation and Definitions Motivation Equivalence relations Some results (Ortiz and Serna) Diagram of Properties (Ortiz and Serna) � FFD UFD BFD ACCP atomic ∗ τ -FFD ∗ ∗ ∗ � τ -ACCP � τ -atomic τ -UFD τ -BFD τ -HFD Figure: In this case τ is an associated-preserving multiplicative equivalence relation. R. M. Ortiz-Albino τ -Factorization

Notation and Definitions Motivation Equivalence relations Some results (Ortiz and Serna) Associated-preserving clousure of an equivalence relation Definition Let τ be an equivalence relation on D ♯ . The associated-preserving clousure of τ is denoted by τ ′ , which is the intersection of all associated-preserving equivalence relations on D ♯ containing τ . Theorem Suppose τ (is unital) has the following property: for any x , y ∈ D ♯ and λ ∈ U ( D ) , if x τ y, then ( λ x ) τ ( λ y ) . Then τ ′ = { ( µ 1 x , µ 2 y ) | ( x , y ) ∈ τ and µ 1 , µ 2 ∈ U ( D ) } R. M. Ortiz-Albino τ -Factorization