Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
An Introduction to the Theory of
Non-Unique Factorization in Integral Domains
and Monoids
Scott Chapman
Sam Houston State University
November 21, 2008
Numerical
Monoids
Chapman
Introduction to Factorization Theory
An Example in Almost Every Abstract Algebra
Book
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
√
Let D = Z[ −5]. In D
6 = 2 · 3 = (1 +
√
−5)(1 −
√
−5)
represents a nonunique factorization into products of
irreducible elements.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
An Example in Almost Every Abstract Algebra
Book
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
√
Let D = Z[ −5]. In D
6 = 2 · 3 = (1 +
√
−5)(1 −
√
−5)
represents a nonunique factorization into products of
irreducible elements.
Let’s consider this situation more closely.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Ideal Theory Behind the Example
Introduction
to
Factorization
Theory
Chapman
√
In Z[ −5] we have the following ideal decompositions:
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Ideal Theory Behind the Example
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
√
In Z[ −5] we have the following ideal decompositions:
h2i
= h2, 1 +
h3i
= h3, 1 −
h1 +
h1 −
√
√
−5i = h2, 1 +
−5i = h2, 1 +
√
√
√
√
−5i2
−5ih3, 1 +
−5ih3, 1 +
−5ih3, 1 −
√
√
√
−5i
−5i
−5i
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Ideal Theory Behind the Example
Introduction
to
Factorization
Theory
And so
Chapman
Introduction
and
Motivation
h6i = h2ih3i =
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Ideal Theory Behind the Example
Introduction
to
Factorization
Theory
And so
Chapman
h6i = h2ih3i =
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
h2, 1 +
√
−5i2 h3, 1 −
√
−5ih3, 1 +
√
−5i =
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Ideal Theory Behind the Example
Introduction
to
Factorization
Theory
And so
Chapman
h6i = h2ih3i =
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
h2, 1 +
h2, 1 +
√
√
−5i2 h3, 1 −
−5ih3, 1 +
√
√
−5ih3, 1 +
−5ih2, 1 +
√
√
−5i =
−5ih3, 1 −
Numerical
Monoids
Chapman
Introduction to Factorization Theory
√
−5i =
Ideal Theory Behind the Example
Introduction
to
Factorization
Theory
And so
Chapman
h6i = h2ih3i =
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
h2, 1 +
h2, 1 +
√
√
−5i2 h3, 1 −
−5ih3, 1 +
√
h1 +
Chapman
√
−5ih3, 1 +
−5ih2, 1 +
√
√
−5i =
−5ih3, 1 −
√
√
−5ih1 − −5i
Introduction to Factorization Theory
√
−5i =
What is Happening in General?
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
Let K = Q(α) be an algebraic extension of Q and OK its ring
of integers.
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
What is Happening in General?
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Let K = Q(α) be an algebraic extension of Q and OK its ring
of integers.
OK is a Dedekind domain and hence each fractional idea of
OK factors uniquely into a product of prime ideals.
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
What is Happening in General?
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Let K = Q(α) be an algebraic extension of Q and OK its ring
of integers.
OK is a Dedekind domain and hence each fractional idea of
OK factors uniquely into a product of prime ideals.
Let Cl(OK ) represent the ideal class group of OK .
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
What is Happening in General?
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Let K = Q(α) be an algebraic extension of Q and OK its ring
of integers.
OK is a Dedekind domain and hence each fractional idea of
OK factors uniquely into a product of prime ideals.
Let Cl(OK ) represent the ideal class group of OK .
Hence, the class number of Q(α) is | Cl(OK ) |.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Well Known Result from Algebraic Number Theory
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
Proposition
The Language
of
Non-Unique
Factorizations
OK has unique factorization of elements into products of
irreducibles (i.e., OK is a UFD) if and only if | Cl(OK ) |≤ 1.
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Famous Result of Carlitz (PAMS 1960)
Introduction
to
Factorization
Theory
Do higher order class numbers have an arithmetic
interpretation?
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Famous Result of Carlitz (PAMS 1960)
Introduction
to
Factorization
Theory
Do higher order class numbers have an arithmetic
interpretation?
Chapman
Theorem
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
The algebraic number field K = Q(α) has class number ≤ 2 if
and only if for every nonzero integer x ∈ OK the number of
primes πj in every factorization
x = π1 π2 · · · πk
Block
Monoids
Types of
Monoids
Examined in
the Literature
only depends on x.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Famous Result of Carlitz (PAMS 1960)
Introduction
to
Factorization
Theory
Do higher order class numbers have an arithmetic
interpretation?
Chapman
Theorem
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
The algebraic number field K = Q(α) has class number ≤ 2 if
and only if for every nonzero integer x ∈ OK the number of
primes πj in every factorization
x = π1 π2 · · · πk
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
only depends on x.
In general, an integral domain with this property (i. e., every
irreducible factorization of a given element has the same
length) is known as a half-factorial domain.
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Let
M = commutative cancellative monoid
written multiplicatively with identity element 1 and associated
group of units M × . Set M ∗ = M\M × .
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Let
M = commutative cancellative monoid
written multiplicatively with identity element 1 and associated
group of units M × . Set M ∗ = M\M × .
We use the usual conventions involving divisibility:
x | y in M ⇐⇒ xz = y for some z ∈ M.
If x | y and y | x in M, then x and y are associates.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Call x ∈ M ∗
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
(1) prime if whenever x | yz for x, y , and z in M, then either
x | y or x | z.
(2) irreducible (or an atom) if whenever x = yz for x, y , and
z in M, then either y ∈ M × or z ∈ M × .
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Call x ∈ M ∗
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
(1) prime if whenever x | yz for x, y , and z in M, then either
x | y or x | z.
(2) irreducible (or an atom) if whenever x = yz for x, y , and
z in M, then either y ∈ M × or z ∈ M × .
As usual,
x prime in M =⇒ x irreducible in M
but not conversely.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Chapman
Set
A(M) = the set of irreducibles of M
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
and
P(M) = the set of prime elements of M.
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Chapman
Set
A(M) = the set of irreducibles of M
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
and
P(M) = the set of prime elements of M.
If M ∗ = hA(M)i, then M is called atomic.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Some Notation
Introduction
to
Factorization
Theory
Chapman
Set
A(M) = the set of irreducibles of M
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
and
P(M) = the set of prime elements of M.
Block
Monoids
If M ∗ = hA(M)i, then M is called atomic.
Types of
Monoids
Examined in
the Literature
If D is an integral domain, then we can apply these conventions
to D by setting M = D • .
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Sets of Lengths
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
For x ∈ M ∗ , the set
L(x) = {n | x = x1 , . . . , xn with each xi ∈ A(M)}
is called the set of lengths of factorizations of x and
L(M) = {L(x) | x ∈ M ∗ }
is called the set of lengths of M.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Sets of Lengths
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
For x ∈ M ∗ , the set
L(x) = {n | x = x1 , . . . , xn with each xi ∈ A(M)}
is called the set of lengths of factorizations of x and
L(M) = {L(x) | x ∈ M ∗ }
is called the set of lengths of M.
Theorem (Geroldinger 1988)
If R is a ring of integers in a finite extension K of Q and
x ∈ R • , then L(x) is an almost arithmetic multiprogression.
Chapman
Introduction to Factorization Theory
Sets of Lengths
Introduction
to
Factorization
Theory
Chapman
What this essentially means is that for x ∈ R • there exists a set
of finite arithmetic sequences A1 , . . . , At and positive integers
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
x1 < x2 < . . . < xs and z1 < z2 < . . . < zr
such that
L(x) = {x1 , . . . , xs } ∪
t
[
At ∪ {z1 , . . . , zr }.
i=1
with xs < min
St
i=1 At
and max
St
i=1 At
< z1 .
Numerical
Monoids
Chapman
Introduction to Factorization Theory
On the Elasticity
Introduction
to
Factorization
Theory
Let M be an atomic monoid. Define for x ∈ M ∗
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
L(x) = sup L(x) and l(x) = inf L(x),
and
ρ(x) =
L(x)
l(x)
to be their quotient. ρ(x) is called the elasticity of x.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
On the Elasticity
Introduction
to
Factorization
Theory
Let M be an atomic monoid. Define for x ∈ M ∗
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
L(x) = sup L(x) and l(x) = inf L(x),
and
ρ(x) =
L(x)
l(x)
to be their quotient. ρ(x) is called the elasticity of x.
We also define
ρ(M) = sup{ρ(x) | x ∈ M ∗ }.
ρ(M) is called the elasticity of M.
Chapman
Introduction to Factorization Theory
Variations on the Elasticity
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
If
{ρ(x) | x ∈ M ∗ } = [1, ρ(M)] ∩ Q,
then M is called fully elastic.
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Variations on the Elasticity
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
If
{ρ(x) | x ∈ M ∗ } = [1, ρ(M)] ∩ Q,
then M is called fully elastic.
If there exists an x ∈ M ∗ such that
Block
Monoids
Types of
Monoids
Examined in
the Literature
ρ(M) = ρ(x)
then the elasticity of M is said to be accepted.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Elasticity and Rings of Integers
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Theorem (Valenza, J. Number Theory 1990 and others)
If R is the ring of integers in a finite extension of Q, then
ρ(R) =
D(Cl(R))
2
and this elasticity is accepted. Moreover, R is fully elastic.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Elasticity and Rings of Integers
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Theorem (Valenza, J. Number Theory 1990 and others)
If R is the ring of integers in a finite extension of Q, then
ρ(R) =
D(Cl(R))
2
and this elasticity is accepted. Moreover, R is fully elastic.
If G is a finite abelian group, then D(G ) represents the
Davenport Constant of G .
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A General Result on Elasticity
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
Theorem (Anderson-Anderson-Chapman-Smith PAMS 1993)
The Language
of
Non-Unique
Factorizations
Let M be a finitely generated commutative cancellative atomic
monoid. Then ρ(M) is both rational and accepted.
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A Note on Accepted Elasticity
Introduction
to
Factorization
Theory
Chapman
All monoids in the remainder of this talk will have accepted
elasticity. Here is an example of a monoid that does not.
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A Note on Accepted Elasticity
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
All monoids in the remainder of this talk will have accepted
elasticity. Here is an example of a monoid that does not.
Let M = {4, 10, 16, 22, 28, . . .} ∪ {1} = 4 + 6N0 .
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A Note on Accepted Elasticity
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
All monoids in the remainder of this talk will have accepted
elasticity. Here is an example of a monoid that does not.
Let M = {4, 10, 16, 22, 28, . . .} ∪ {1} = 4 + 6N0 .
Facts: (proofs are homework for your algebra/number theory
classes)
ρ(M) = 2.
For each x > 1 in M, ρ(x) < 2.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A Note on Accepted Elasticity
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
All monoids in the remainder of this talk will have accepted
elasticity. Here is an example of a monoid that does not.
Let M = {4, 10, 16, 22, 28, . . .} ∪ {1} = 4 + 6N0 .
Facts: (proofs are homework for your algebra/number theory
classes)
ρ(M) = 2.
For each x > 1 in M, ρ(x) < 2.
Hint: The atoms of M come in two forms, those exactly
divisible by 2 or those exactly divisible by 4.
Chapman
Introduction to Factorization Theory
The Difference Set
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Let M be an atomic monoid and x ∈ M ∗ . Suppose
L(x) = {x1 , x2 , . . . , xt }
with x1 < x2 < . . . < xt . Set
∆(x) = {xi − xi−1 | 2 ≤ i ≤ t}.
We call ∆(x) the difference set of x.
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
The Difference Set
Introduction
to
Factorization
Theory
Let M be an atomic monoid and x ∈ M ∗ . Suppose
Chapman
Introduction
and
Motivation
L(x) = {x1 , x2 , . . . , xt }
with x1 < x2 < . . . < xt . Set
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
∆(x) = {xi − xi−1 | 2 ≤ i ≤ t}.
We call ∆(x) the difference set of x.
Set
∆(M) =
[
∆(x).
x∈M ∗
We call ∆(M) the difference set of M.
Chapman
Introduction to Factorization Theory
Basic Difference Set Result
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Theorem (Geroldinger 1991)
If M is a reduced atomic monoid, then
min ∆(M) = gcd ∆(M).
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Basic Difference Set Result
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Theorem (Geroldinger 1991)
If M is a reduced atomic monoid, then
min ∆(M) = gcd ∆(M).
Hence, if d = gcd ∆(M), then
{d} ⊆ ∆(M) ⊆ {d, 2d, . . . , td}
for some nonnegative integer t.
Chapman
Introduction to Factorization Theory
The Delta Set of a Ring of Integers
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Theorem (Geroldinger ≈ 1992)
Let R be the ring of integers in a finite extension of Q. Then
∆(R) ⊆ {1, 2, . . . , D(Cl(R)) − 2}.
If the class number of R is a prime p, then
∆(R) = {1, 2, . . . , p − 2}.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Definition of a Block Monoid
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Let G be an abelian group and
Y n
F(G ) = {
gi i | only finitely many ni 6= 0}
gi ∈G
be the free abelian monoid on G .
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Definition of a Block Monoid
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Let G be an abelian group and
Y n
F(G ) = {
gi i | only finitely many ni 6= 0}
gi ∈G
be the free abelian monoid on G .
Let
B(G ) = {
Y
gi ∈G
gini |
X
ni gi = 0}
gi ∈G
be the submonoid of F(G ) consisting of zero-sum sequences.
B(G ) is known as the block monoid on G .
Chapman
Introduction to Factorization Theory
An Example
Introduction
to
Factorization
Theory
Let Z3 = {0̄, 1̄, 2̄}. Then
Chapman
Introduction
and
Motivation
B(Z3 ) = {0̄n0 1̄n1 2̄n2 | n1 + 2n2 ≡ 0
(mod 3)}.
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
An Example
Introduction
to
Factorization
Theory
Let Z3 = {0̄, 1̄, 2̄}. Then
Chapman
B(Z3 ) = {0̄n0 1̄n1 2̄n2 | n1 + 2n2 ≡ 0
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
(mod 3)}.
Notice that the irreducible elements of B(Z3 ) are
0̄,
1̄3 ,
2̄3 , and
1̄2̄.
Chapman
Introduction to Factorization Theory
The Connection with Rings of Integers
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Let R be a ring of integers in a finite extension of Q and
α ∈ R • . Suppose
hαi = P1 · · · Pk
for not necessarily distinct prime ideals of R. Let [Pi ] represent
the image of the ideal Pi in Cl(R).
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
The Connection with Rings of Integers
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Let R be a ring of integers in a finite extension of Q and
α ∈ R • . Suppose
hαi = P1 · · · Pk
for not necessarily distinct prime ideals of R. Let [Pi ] represent
the image of the ideal Pi in Cl(R).
Define
Block
Monoids
Types of
Monoids
Examined in
the Literature
ϕ : R • → B(Cl(R))
by
ϕ(α) = [P1 ] · · · [Pk ].
Numerical
Monoids
Chapman
Introduction to Factorization Theory
The Connection with Rings of Integers
Introduction
to
Factorization
Theory
Chapman
Facts:
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
ϕ is a monoid homomorphism from R • to B(Cl(R)).
L(α) = L([P1 ] · · · [Pk ]).
Moreover, {L(α) | α ∈ R • } = {L(B) | B ∈ B(Cl(R))}.
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
The Connection with Rings of Integers
Introduction
to
Factorization
Theory
Chapman
Facts:
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
ϕ is a monoid homomorphism from R • to B(Cl(R)).
L(α) = L([P1 ] · · · [Pk ]).
Moreover, {L(α) | α ∈ R • } = {L(B) | B ∈ B(Cl(R))}.
Moral: All the results so far for rings of integers are
specializations of more general results on Block Monoids.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A Final Word on Block Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Question: If G1 and G2 are finite abelian groups and
L(B(G1 )) = L(B(G2 ))
then must G1 ∼
= G2 .
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
A Final Word on Block Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Question: If G1 and G2 are finite abelian groups and
L(B(G1 )) = L(B(G2 ))
then must G1 ∼
= G2 .
Answer: NO, but very few exceptional cases are known
(L(Z3 ) = L(Z2 ⊕ Z2 ) is one).
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Types of Monoids Examined in the Literature
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
Block Monoids (Israel Math. J. 1990)
Semigroup Rings (Archiv Math. 1993)
The Language
of
Non-Unique
Factorizations
Diophantine Monoids (Pacific J. Math. 2002)
Block
Monoids
The Ring of Integer-Valued Polynomials (J. Algebra 2005)
Types of
Monoids
Examined in
the Literature
Congruence Monoids (Colloq. Math. 2007)
Numerical Monoids (rest of talk)
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Definition of a Numerical Monoid
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
If n1 < n2 < · · · < nt are natural numbers, then set
hn1 , n2 , . . . , nt i =
{x1 n1 + x2 n2 + · · · + xt nt | each xi ∈ N0 }.
hn1 , n2 , . . . , nt i is an additive submonoid of N0 .
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Definition of a Numerical Monoid
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
If n1 < n2 < · · · < nt are natural numbers, then set
hn1 , n2 , . . . , nt i =
{x1 n1 + x2 n2 + · · · + xt nt | each xi ∈ N0 }.
hn1 , n2 , . . . , nt i is an additive submonoid of N0 .
Such a submonoid is known as a numerical monoid. The
integers n1 , . . . , nt are called the generators of hn1 , n2 , . . . , nt i
and if they are relatively prime, then hn1 , n2 , . . . , nt i is called a
primitive numerical monoid.
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Definition of a Numerical Monoid
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
If n1 < n2 < · · · < nt are natural numbers, then set
hn1 , n2 , . . . , nt i =
{x1 n1 + x2 n2 + · · · + xt nt | each xi ∈ N0 }.
hn1 , n2 , . . . , nt i is an additive submonoid of N0 .
Such a submonoid is known as a numerical monoid. The
integers n1 , . . . , nt are called the generators of hn1 , n2 , . . . , nt i
and if they are relatively prime, then hn1 , n2 , . . . , nt i is called a
primitive numerical monoid.
Simple Fact: Every numerical monoid is isomorphic to a
primitive numerical monoid.
Chapman
Introduction to Factorization Theory
Notes on Numerical Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
Let S = hn1 , . . . , nt i be a primitive numerical monoid. The
largest n ∈ N such that n 6∈ S is called the Frobenius Number
of S and denoted F(S).
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Notes on Numerical Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Let S = hn1 , . . . , nt i be a primitive numerical monoid. The
largest n ∈ N such that n 6∈ S is called the Frobenius Number
of S and denoted F(S).
Theorem (Sylvester 1884)
Let S = hn1 , n2 i be a primitive numerical monoid. Then
F(S) = (n1 − 1)(n2 − 1) − 1 = n1 n2 − n1 − n2 .
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Elastic Properties of Numerical Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Theorem (2002 Trinity REU - Rocky Mountain J. Math. 2006)
Let S = hn1 , n2 , . . . , nt i be a primitive numerical monoid whose
given generating set is minimal with t ≥ 2.
ρ(S) =
nt
n1 .
There exists a rational number q with 1 < q < nn1t such
that ρ(s) 6= q for all s ∈ S (i.e., S is not fully elastic)
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Delta Sets of Numerical Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Theorem (Trinity REU 2004 - J. Algebra Appl. 2006)
If S = hn1 , . . . , nk i then
min ∆(S) = gcd{ni − ni−1 | 2 ≤ i ≤ k}.
If S = hn, n + k, n + 2k, . . . , n + tki, then ∆(S) = {k}.
For any positive integers k and t, there exists a three
generated numerical monoid S such that
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
∆(S) = {k, 2k, . . . , tk}.
For n ≥ 3,
∆(hn, n + 1, n2 − n − 1i) = {1, 2, .., n − 2} ∪ {2n − 5}.
Chapman
Introduction to Factorization Theory
Length Sets and Numerical Monoids
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Theorem (Trinity REU 2006 - Integers 2007)
Let S and S 0 be numerical monoids requiring at least two
generators. Then L(S) = L(S 0 ) does not imply that S = S 0 .
In fact, set
S = ha, a + k, . . . , a + wki,
with 0 ≤ w < a and gcd(a, k) = 1, and also
S 0 = hc, c + t, . . . , c + vti,
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
where v < c, gcd(c, t) = 1. If S 6= S 0 , then the following
statements are equivalent:
L(S) = L(S 0 )
k = t,
c
a
=
v
w,
gcd(a, w ) ≥ 2, and gcd(c, v ) ≥ 2.
Chapman
Introduction to Factorization Theory
Periodic Delta Sets
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
Theorem (Trinity REU 2007 - to appear in Aequationes Math)
Given a primitive numerical monoid M = hn1 , . . . , nk i, we have
for all x ≥ 2kn2 nk2 that ∆(x) = ∆(x + n1 nk ).
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Numerical
Monoids
Chapman
Introduction to Factorization Theory
Periodic Delta Sets
Introduction
to
Factorization
Theory
Chapman
Introduction
and
Motivation
The Language
of
Non-Unique
Factorizations
Block
Monoids
Types of
Monoids
Examined in
the Literature
Theorem (Trinity REU 2007 - to appear in Aequationes Math)
Given a primitive numerical monoid M = hn1 , . . . , nk i, we have
for all x ≥ 2kn2 nk2 that ∆(x) = ∆(x + n1 nk ).
Corollary
Let M = hn1 , . . . , nk i be a primitive numerical monoid, then if
we set N = 2kn2 nk2 + n1 nk , we have:
∆(M) =
[
∆(x)
x∈M,x<N
Numerical
Monoids
Chapman
Introduction to Factorization Theory