FACTORIZATIONS OF UPPER TRIANGULAR MATRICES WITH
NON-COMMUTATIVE ENTRIES
by
Joel Je↵ries
An Abstract
presented in partial fulfillment
of the requirements for the degree of
Master of Science
in the Department of Mathematics and Computer Science
University of Central Missouri
May, 2016
ABSTRACT
by
Joel Je↵ries
The study of factorization theory concerns writing elements of various algebraic structures
as products of other unfactorable elements. Factorization in commutative semigroups has
been well studied, but far less work has been done in the non-commutative setting. We
consider the factorization of triangular matrices with non-commutative entries under usual
matrix multiplication. In particular, we provide a construction of a semigroup of block
triangular matrices with diagonal entries from matrix semigroups of arbitrary sizes but over
the same underlying ring. We then relate factorization properties in this semigroup to easierto-understand semigroups, in particular products of semigroups, by means of a new tool, the
weak transfer homomorphism.
FACTORIZATIONS OF UPPER TRIANGULAR MATRICES WITH
NON-COMMUTATIVE ENTRIES
by
Joel Je↵ries
A Thesis
presented in partial fulfillment
of the requirements for the degree of
Master of Science
in the Department of Mathematics and Computer Science
University of Central Missouri
May, 2016
FACTORIZATIONS OF UPPER TRIANGULAR MATRICES WITH
NON-COMMUTATIVE ENTRIES
by
Joel Je↵ries
APPROVED:
Thesis Chair
Thesis Committee Member
Thesis Committee Member
ACCEPTED:
Chair, Department of Mathematics
and Computer Science
UNIVERSITY OF CENTRAL MISSOURI
WARRENSBURG, MISSOURI
ACKNOWLEDGMENTS
I would first like to thank my family for always pushing me to do the best I can and for
providing support whenever I am in need. I would especially like to thank my wife, Liz, for
supporting me in all of our journeys, wherever they take us. Next, I would like to thank
my advisor, Dr. Nicholas Baeth, for his help with this thesis and many of my academic
endeavors. I would also like to thank my readers, Dr. Rhonda McKee and Dr. Mahmound
Yousef, for helping me improve and revise my thesis. Lastly, I would like to thank everyone
at the math department at UCM for helping me become an excellent mathematician and
teacher.
Contents
1 Definitions and Foundation
1
2 Weak Transfer Homomorphisms
12
3 Direct Products of Semigroups
20
4 Generalized Block Triangular Matrices
31
5 Factoring Block Triangular Matrices
40
6 Applications and Examples
47
Bibliography
59
vi
Chapter 1
Definitions and Foundation
The study of factorization in rings and semigroups has been a growing field of research over
the past few decades. One of the major concerns in factorization theory is the study of how
elements of various algebraic structures can be written as products of other non-factorable
elements. This is done by studying invariants of factorizations, such as the lengths of such
products.
Much of this study has been restricted to the commutative and cancellative setting (see
[GHK06]). However, some recent work has been done to generalize results from these commutative and cancellative settings to non-cancellative or non-commutative settings. For example, Adams et. al. [DA11] study semigroups of matrices with integer entries, Geroldinger
[Ger13] studies Krull rings, a well-behaved class of non-commutative rings, and Smertnig
[Sme13] considers even more general non-commutative settings. In many of these studies,
such as [Sme13] and [Ger13], homomorphisms were constructed from the non-commutative
structures to more familiar commutative structures, allowing an easier way to study factorizations. The work of Baeth and Smertnig, in [BS15], extends some of these results to more
general non-commutative settings. In 2013, Bachman, Baeth and Gossel [DB14] considered
cancellative semigroups of upper triangular matrices with entries from integral domains and
1
CHAPTER 1. DEFINITIONS AND FOUNDATION
2
were able to characterize factorizations of these these matrices by studying factorizations
of their diagonal elements in the underlying integral domain. In this thesis we focus on
non-commutative but cancellative structures. We extend the results in [DB14] to triangular
matrices with a non-commutative underlying ring, as well as to block triangular matrices.
In the remaining parts of this chapter, we give definitions and examples to provide background for the rest of the thesis. In Chapter 2, we introduce the weak transfer homomorphism, a useful tool in the study of factorizations. Chapter 3 concerns some of the properties
of factorization in direct products of semigroups. In addition, we consider how weak transfer
homomorphisms can be of use in the study of factorization in direct products of semigroups.
In Chapter 4, we introduce the concept of generalized triangular block matrices and discuss
their properties. Our main result, Theorem 5.2, is shown in Chapter 5, in which we apply a
weak transfer homomorphism from this defined set of matrices to a product space. Finally,
in Chapter 6, we discuss the implications of the main result and include some examples in
which it can be applied.
First, we introduce some notation and basic definitions.
Notation 1.1.
1. We take N = {1, 2, . . . } and N0 = {0, 1, 2, . . . }.
2. We use the discrete interval notation, where if a, b 2 N with a < b, then
[a, b] = {a, a + 1, . . . , b
1, b}.
Definition 1.2. A semigroup (S, ·) is a set S together with an associative cancellative
operation · : S ⇥ S ! S where ·(s1 , s2 ) is denoted as s1 · s2 .
Note that the usual definition of a semigroup need not have a cancellative operation.
However, the absence of this condition introduces many complications to the study of factorization. For the sake of this thesis, we only consider semigroups with a cancellative
CHAPTER 1. DEFINITIONS AND FOUNDATION
3
operation. Furthermore, throughout, we will write semigroups as S instead of (S, ·) when
the operation is understood and will write ab instead of a · b.
Example 1.3. The following are examples of semigroups:
• The set of non-zero elements of any cancellative ring, such as Z or Z[x3 , x4 , x5 ]. We
denote the non-zero elements of a cancellative ring R by R• .
• For any cancellative ring R, the triangular matrices over R• .
Definition 1.4. We say a semigroup S is commutative if ab = ba for all a, b 2 S.
Next we define some properties of semigroups and their elements that are essential in the
study of factorization.
Definition 1.5. Let S be a semigroup.
1. An element i 2 S is said to be the identity of S if for all s 2 S, si = is = s. A
semigroup S is said to be a semigroup with unity (or monoid ) if it contains an identity
element. Throughout, we will often use 1 (or I in the case of matrix semigroups) to
represent the identity element of a semigroup.
2. If S has an identity 1, an element u 2 S is called a unit of S if there exists an
element u
1
2 S such that uu
1
= 1 = u 1 u. The set of units of S is the set
S ⇥ = {u : u is a unit of S}. If s 2 S and there is an element x 2 S such that xs = i,
x is called the left inverse of s. If there exists an element y 2 S such that sy = i, y is
called the right inverse of s. Therefore, a unit is an element that has both a left and
right inverse. If a, b 2 S, we call a and b associates if and only if there exists u, v 2 S ⇥
such that a = ubv.
3. An element a is an atom (or irreducible element) of S if whenever a = bc for some
elements b, c 2 S, exactly one of b or c is a unit of S. The semigroup S is said to be
CHAPTER 1. DEFINITIONS AND FOUNDATION
4
atomic provided that every nonunit in S can be written as a finite product of atoms
in S.
4. Let a be a nonunit of S. We denote the set of lengths of a by
LS (a) = {m 2 N : a = a1 · · · am with each ai an atom of S}.
A semigroup is called half-factorial if the length set of every nonunit has cardinality
one.
5. Assume that S is commutative. For any elements a, b 2 S, we say that a divides b
(denoted a|b) if there exists an element x 2 S such that ax = b.
Example 1.6. The following are some examples of the definitions from Definition 1.5 in the
semigroup P = Z[x3 , x4 , x5 ]• .
• 1 is the identity of P.
• P ⇥ = {1, 1}.
• x3 , x4 , and x5 are all atoms of P.
• x3 and
x3 are associates.
• LP (x4 ) = {1} since x4 is an atom. LP (1) = {0} since 1 is a unit.
5
2
3
3
• LP (x15 ) = {3, 4, 5} since x15 = (x3 ) = (x3 ) x4 x5 = (x3 )(x4 ) = (x5 ) are the only
factorizations of x15 as a product of atoms, up to associates and permutations.
We now define several classes of semigroups, each of which have di↵erent properties that
a↵ect the factorization of their elements.
CHAPTER 1. DEFINITIONS AND FOUNDATION
5
Definition 1.7.
1. An integral domain (D, +, ·) is a commutative ring with identity 1 such that for all
a, b 2 D, if ab = 0 then either a = 0 or b = 0. Note that (D• \{0}, ·) is a semigroup in
accordance with Definition 1.2.
2. A unique factorization domain D is an integral domain in which every nonunit of D can
be written as a product of irreducible elements, uniquely up to order and associates.
It is well known that all atoms of a unique factorization domain are prime.
3. Let D be a domain. An ideal in D is a subset I of elements in D such that (I, +) is a
group and for all d 2 D, x 2 I, we have dx 2 I and xd 2 I.
4. A principal ideal domain is an integral domain in which every ideal can be generated
by a single element. We note that every principal ideal domain is a unique factorization
domain.
5. A semigroup S with unity is called Dedekind-finite if for any u, v 2 S, uv = 1 implies
vu = 1. Note that all commutative semigroups are trivially Dedekind-finite. Also note
that in Dedekind-finite semigroups, if uv = 1, u, v 2 S ⇥ .
6. A ring R is said to be stably finite if, for all square matrices A and B of the same size
over R, AB = 1 implies BA = 1. That is, R is stably finite if and only if the semigroup
of n ⇥ n matrices over R that are not zero-divisors is Dedekind-finite for all n 2 N.
Equivalently, for any R-module K, Rn
K⇠
= Rn implies K = 0.
It is nice to note that any commutative semigroup is Dedekind-finite. Also, it is shown
in [Lam99, Proposition 6.60] that any Noetherian ring is Dedekind-finite and, in [GHK06],
any Noetherian ring is atomic. Therefore, Noetherian rings hold many of the properties
necessary for the study of factorization in this thesis.
CHAPTER 1. DEFINITIONS AND FOUNDATION
6
The next few lemmas provide useful facts about the product of units and atoms of a
semigroup. We will use these lemmas throughout the thesis.
Lemma 1.8. Let S be a semigroup with unity and n 2 N.
1. If u1 , . . . , un 2 S ⇥ , then u1 · · · un 2 S ⇥ .
2. If S is Dedekind-finite, then u1 , . . . , un 2 S ⇥ if and only if u1 · · · un 2 S ⇥ .
Proof. First, we prove (1) by induction. Let u, v 2 S ⇥ . Then there exists u
uu
1
= 1 = u 1 u. Similarly, since v 2 S ⇥ , there exists v
Then (uv)(v 1 u 1 ) = u(vv 1 )u
Since v 1 u
1
1
= uu
1
1
2 S ⇥ such that
2 S ⇥ such that vv
1
= 1 = v 1 v.
= 1 and (v 1 u 1 )(uv) = v 1 (u 1 u)v = v 1 v = 1.
2 S, uv 2 S ⇥ . Now assume (1) holds for a product of n
2 S ⇥ , u1 · · · u n
1
1 units. That is, if
2 S ⇥ . Let u1 , . . . , un 2 S ⇥ . Then by the inductive hypothesis,
u1 , . . . , u n
1
u1 · · · un
2 S ⇥ . Thus, as proven in the case n = 2, u1 · · · un = (u1 · · · un 1 )un 2 S ⇥ .
1
1
To prove (2), we need only prove the converse of this statement in the case of Dedekindfinite semigroups. We do so by induction. Suppose u1 u2 2 S ⇥ . Then there exists v 2 S ⇥ such
that (u1 u2 )v = 1 = v(u1 u2 ). Then u1 (u2 v) = 1 and, since S is Dedekind-finite, (u2 v)u1 = 1
and u1 2 S ⇥ . Similarly, (vu1 )u2 = 1, and thus u2 (vu1 ) = 1 and u2 2 S ⇥ . Now suppose that
the result holds for the product of n
each i 2 [1, n
1 elements. That is, if u1 · · · un
1]. Then if u1 · · · un = (u1 · · · un 1 )un 2 S ⇥ , u1 · · · un
1
1
2 S ⇥ , ui 2 S ⇥ for
2 S ⇥ and un 2 S ⇥
since we have shown that if the product of two elements is a unit, each is a unit. Thus, by
our inductive hypothesis, since u1 · · · un
1
2 S ⇥ , ui 2 S ⇥ for all i 2 [1, n
1]. Thus, ui 2 S ⇥
for all i 2 [1, n].
The converse of statement (1) in Lemma 1.8 is not necessarily true in non-Dedekindfinite semigroups. We provide an example of such a non-Dedekind-finite semigroup and the
problems such a property creates in the study of factorization.
CHAPTER 1. DEFINITIONS AND FOUNDATION
7
Example 1.9. Let T be the ring of linear transformations on infinite sequences with i
the identity transformation. Let L be the left-shift transformation where L((x1 , x2 , . . . )) =
(x2 , x3 , . . . ) and let R be the right-shift function, where R((x1 , x2 , . . . )) = (0, x1 , x2 , . . . ).
Then LR((x1 , x2 , . . . )) = (x1 , x2 , . . . ) for all infinite sequences, and so LR = i. However,
RL((x1 , x2 , . . . )) = (0, x2 , x3 , . . . ), which is not an idenity transformation unless x1 = 0, and
so RL 6= i. Thus, the identity i, a unit of T • , can be written as a product of two nonunits.
Furthermore, L is not injective and R is not surjective, so neither are invertible.
The next lemma provides a result similar to Lemma 1.8, but for products of units and
atoms.
Lemma 1.10. Let S be a semigroup with unity, n 2 N, and a1 , . . . , an 2 S.
1. An element associate to an atom is an atom.
2. If there exists h 2 [1, n] such that ah is an atom of S and ai 2 S ⇥ for all i 2 [1, n]\{h},
then a1 · · · an is an atom of S.
3. If S is Dedekind-finite, a1 · · · an is an atom of S if and only if there is a h 2 [1, n] such
that ah is an atom of S and ai 2 S ⇥ for each i 2 [1, n]\{h}.
Proof. First, we prove (1). Let a = uxv, where u, v 2 S ⇥ and x is an atom of S. Suppose
a = bc for some b, c 2 S. Then x = u 1 (bc)v
u 1 b 2 S ⇥ or cv
1
1
= (u 1 b)(cv 1 ). Since x is an atom, either
2 S ⇥ . Let µ = u 1 b and ⌘ = cv 1 . Suppose µ 2 S ⇥ . Then b = uµ and,
by Lemma 1.8, b 2 S ⇥ since it is the product of units. Alternatively, suppose ⌘ 2 S ⇥ . Then
c = ⌘v and, by Lemma 1.8, c 2 S ⇥ . Therefore, a is an atom of S.
We use (1) to prove (2). Let a1 , . . . , an 2 S and h 2 [1, n] such that ah is an atom of
S and ai 2 S ⇥ for i 2 [1, n]\{h}. Then a = a1 · · · an = (a1 · · · ah 1 )ah (ah+1 . . . an ) = uah v,
where u = a1 · · · ah
1
and v = ah+1 . . . an . By Lemma 1.8, u, v 2 S ⇥ since each is the product
of units. Thus, by (1), a is an atom of S.
CHAPTER 1. DEFINITIONS AND FOUNDATION
8
To prove (3), we need only prove the converse of (2) in the case of Dedekind-finite
semigroups. We do so by induction. Suppose a = a1 a2 is an atom. Then, by the definition
of an atom, either a1 or a2 is a unit. If a1 2 S ⇥ , then a2 = a1 1 a, and thus by (1), a2 is an
atom of S. Alternatively, if a2 2 S ⇥ , then a1 = aa2 1 , and thus by (1), a1 is an atom of S.
Thus, if n = 2, the result holds. Now suppose that the result holds for the product of n
elements. That is, if a1 · · · an
1
is an atom of S then there is a h 2 [1, n
an atom of S and ai 2 S ⇥ for i 2 [1, n
1] such that ah is
1]\{h}. Suppose a = a1 · · · an is an atom of S. Then
a = (a1 · · · an 1 )an . Hence, from the case n = 2, either an is an atom and a1 · · · an
or a1 · · · an
1
is an atom and an 2 S ⇥ . If a1 · · · an
for each i 2 [1, n
exists h 2 [1, n
1]. If a1 · · · an
1
1
1
1
2 S⇥
2 S ⇥ , then by Lemma 1.8, ai 2 S ⇥
is an atom, then by the inductive hypothesis, there
1] such that ah is an atom of S and ai 2 S ⇥ for each i 2 [1, n
1]\{h}.
In either case, there exists h 2 [1, n] such that ah is an atom of S and ai 2 S ⇥ for each
i 2 [1, n]\{h}.
Again, the converse of (2) is not generally true in non-Dedekind-finite semigroups. We
provide an example of why this can be difficult in the study of factorization.
Example 1.11. Let T be the ring of linear transformations on infinite sequences as described
in Example 1.9. Let A be any element of T . Then, since i = RL,
A = Ai = A(RL) = A(RL)(RL) = · · · .
Since neither R nor L is a unit, A can be factored into a product of non-units regardless of
whether A is a unit or a non-unit.. Furthermore, A can be written as a product of infinitely
many non-unit elements.
In non-commutative semigroups, units do not necessarily commute with elements other
than their inverse and the identity. This makes it difficult to distinguish atoms in a large
CHAPTER 1. DEFINITIONS AND FOUNDATION
9
product. The following lemma provides a way to write a product of atoms and units as a
product of only atoms. It will be useful in many of the proofs in this thesis.
Lemma 1.12. Let S be an atomic semigroup with unity and n, m, k1 , . . . , km 2 N such that
1  k1 < k2 < · · · < km  n. If x = s1 · · · sn such that sk1 , sk2 , . . . , skm are atoms of S and
si 2 S ⇥ for i 2
/ {k1 , . . . , km }, then there exist atoms t1 , . . . , tm in S such that x = t1 · · · tm
and, for each i 2 [1, m], ti = ui ski vi for some ui , vi 2 S ⇥ .
Proof. Suppose x = s1 · · · sn with sk1 , sk2 , . . . , skm atoms in this product and with all other
Y
Y
terms in the product units of S. Define t1 =
si , t j =
si for j 2 [2, m 1],
and tm =
Y
Y
km in
. Then t1 · · · tm = s1 · · · sn = x. Furthermore, t1 = u1 sk1 v1 , where u1 =
si and v1 =
1i<k1
Y
si , tm = um skm vm , where um = 1 and vm =
k1 <i<k2
j 2 [2, m
kj i<kj +1
1i<k2
1], tj = uj skj vj , where uj = 1 and vj =
Y
Y
si , and for all
km <in
si . Then by Lemma 1.8, for
kj <i<kj+1
all i 2 [1, m], ui and vi are units since each is either a product of units or is the identity.
Clearly, for each i 2 [1, m], ti = ui ski vi for some ui , vi 2 S ⇥ .
The example below illustrates the construction used in Lemma 1.12.
Example 1.13. Consider the semigroup of n ⇥ n matrices over Z that are not zero divisors.
Note that in this semigroup, a matrix is invertible if and only if its determinant is 1 or
1
and that a matrix is irreducible if and only if its determinant is prime (this is discussed in a
more general setting in Remark 2.3). By the construction in Lemma 1.12, we can write the
product below in the following way:
"
#"
#"
#"
#"
#"
#"
#"
#"
#
1 0 0 1 1 2 0 1
3 1
1 0 7 2 0 1 1 0
0 1 1 0 3 4 1 0
5 2 0 1 5 3
1 0 0 1
"
#"
#"
#"
#! "
#"
#! "
#"
#"
#!
1 0 0 1 1 2 0 1
3 1
1 0
7 2 0 1 1 0
=
0 1 1 0 3 4 1 0
5 2 0 1
5 3
1 0 0 1
"
#"
#"
#
4 3
3 1 2 7
=
.
2 1
5 2
3 5
CHAPTER 1. DEFINITIONS AND FOUNDATION
10
Note that in the statement of Lemma 1.12, we write “ti = ui ski vi for some ui , vi 2 S ⇥ ”.
This says that each of the atoms in the resulting product is associate to one of the atoms in
the original product. However, the construction of the proof is a bit stronger, as many of
the atoms only group with units on the right. For ease of notation, we have stated Lemma
1.12 in its weaker form.
We now define two semigroups that will be the main focus of the thesis. In Chapters 5
and 6, we relate these two semigroups and the factorization of their elements.
Definition 1.14.
1. Let n 2 N and let S1 , . . . , Sn be semigroups. We define S1 ⇥ · · · ⇥ Sn =
Cartesian product of the sets S1 , . . . , Sn . That is,
n
Y
n
Y
Si to be the
i=1
Si is the set of all ordered n-tuples
i=1
(s1 , . . . , sn ) such that si 2 Si for each i 2 [1, n]. Define a binary operation · on
n
Y
i=1
component-wise operation. That is, if a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) 2
ab = (a1 b1 , . . . , an bn ). Note that if Si has an identity 1i for each i 2 [1, n],
has the identity (11 , . . . , 1n ). Throughout, we will write the semigroup
simply
n
Y
i=1
Si . If S1 = S2 = · · · = Sn = S, we write
use the bold vector notation to denote elements of
n
Y
i=1
i=1
n
Y
Si ,
i=1
n
Y
i=1
!
Si , ·
Si
as
S = S n . Furthermore we will
i=1
n
Y
n
Y
Si by
Si .
CHAPTER 1. DEFINITIONS AND FOUNDATION
11
2. Let R be a ring with unity. We define Mn (R) to be the semigroup of all non-zerodivisors in the ring of n ⇥ n matrices with entries from R. The operation of this
semigroup is matrix multiplication and the identity of Mn (R) is the identity matrix
In . Similarly, we define Tn (R) to be the semigroup of n ⇥ n upper triangular matrices
with entries from R and with elements from R• along the diagonal. Note that by these
definitions, both Mn (R) and Tn (R) are semigroups. Throughout the thesis, we will use
capital letters to denote matrices.
The restrictions placed on the elements in the defintions of Mn (R) and Tn (R) are to
eliminate matrices that would be zero-divisors. The introduction of such elements make the
study of factorization difficult.
It is nice to note that if R is stably finite, then Mn (R) is Dedekind-finite. Also, as shown
in [Lam99, Section 6F], if R is Noetherian, Mn (R) is Noetherian, and therefore Dedekindfinite.
Chapter 2
Weak Transfer Homomorphisms
Transfer homomorphisms and weak transfer homomorphisms are useful tools in the study
of factorization. They map elements in one semigroup to another while preserving several
properties of factorization, such as length. In this way, these homomorphisms allow one
to study factorizations of elements in a complicated semigroup by instead studying the
factorizations of their images in a simpler semigroup. In this chapter we define and discuss
the transfer homomorphism, which has been traditionally used in factorization theory. We
then generalize the weak transfer homomorphism as introduced in [DB14] to include maps
where both the codomain and domain are non-commutative and as discuss its properties.
Definition 2.1. Let S and T be atomic semigroups with identities with T commutative.
A semigroup homomorphism
: S ! T is a transfer homomorphism if it has the following
properties:
1. T = (S)T ⇥ and
1
(T ⇥ ) = S ⇥
2. Whenever (s) = xy with x and y non-units of T , there exist non-units a and b of S
such that ab = s, (a) = ux, and (b) = vy with u, v 2 T ⇥ .
12
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
In the definition of transfer homomorphisms, property (1) says that
13
is surjective up to
multiplication by a unit. Therefore, if the map is surjective, the property holds. We will
often use this fact. The next theorem is stated and proven in [GHK06] and demonstrates
the use of transfer homomorphisms in factorization theory. We list it here for completeness.
Theorem 2.2. [GHK06, Proposition 3.2.3] Let S and T be atomic semigroups with identities
with T commutative. Let
: S ! T be a transfer homomorphism and let s 2 S. The
following are true:
1. s is an atom of S if and only if (s) is an atom of T
2. LS (s) = LT ( (s))
Remark 2.3. If D is a principal ideal domain, then by using the Smith normal form, it can
be shown ([DA11]) that the determinant map from Mn (D) to D• is a transfer homomorphism.
Example 2.4 demonstrates how transfer homomorphisms can be used to study factorization.
"
#
2 5
2 M2 (Z). Since Z is a principal ideal domain, the
6 9
determinant map is a transfer homomorphism from M2 (Z) to Z• . Thus, since det(A) factors
Example 2.4. Consider A =
uniquely in Z• , up to sign and permutation, as det(A) = 12 = 3(2)2 , LM2 (Z) (A) = {3}. That
is, A can only be written as a product of 3 atoms of M2 (Z). However, even though the
length of any factorization of A is three, these factorizations need not be unique. We list
two di↵erent factorizations below.
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
A =
=
=
"
"
"
2
6
2
4
7
13
14
#
5
9
#"
#"
#
3
3 1
5 2
5 11 3
19 7
#"
#"
#
12
69 95
47 17
.
22 43 61
33 12
Note that each of the six terms of these two products are irreducible since they each have a
prime (and hence irreducible) determinant in Z.
Although transfer homomorphisms are a useful tool in the study of factorization, they do
not always generalize well to non-commutative settings. It is shown in [DB14, Example 4.5]
that there exist semigroups for which there is no transfer homomorphism to any commutative
semigroup. Below we describe a weak transfer homomorphism, a similar tool that can be used
in a non-commutative setting. The weak transfer homomorphism, as introduced in [DB14],
assumes the codomain be commutative. We introduce here the weak transfer homomorphism
in the more general setting where both the domain and codomain may be non-commutative.
Definition 2.5. Let S and T be atomic semigroups with identities. A semigroup homomorphism
: S ! T is a weak transfer homomorphism if it has the following properties:
1. T = T ⇥ (S)T ⇥ and
1
(T ⇥ ) = S ⇥ .
2. Whenever (s) = t1 · · · tn with t1 , . . . , tn atoms of T , there exist s1 , . . . , sn atoms of S
and a permutation
(si ) = ui t
(i) vi ,
: [1, n] ! [1, n] such that s1 · · · sn = s and for each i 2 [1, n],
with each ui and vi a unit of T .
As shown in [DB14, Theorem 3.2], weak transfer homomorphism preserve the properties
of transfer homomorphisms listed in Theorem 2.2 when the codomain T is commutative. We
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
15
give the analogous results when the codomain is not necessarily commutative. The proof is
nearly identical to that found in [DB14].
Theorem 2.6. Let S and T be atomic semigroups with identities. Let
: S ! T be a weak
transfer homomorphism and let s 2 S. The following are true:
1. s is an atom of S if and only if (s) is an atom of T
2. LS (s) = LT ( (s))
Proof. To prove (1), suppose s is an atom of S. Since T is atomic, we may write (s) =
t1 · · · tn with ti an atom of T for each i 2 [1, n]. Then, since
is a weak transfer homomor-
phism, there exist atoms s1 , . . . , sn of S such that s1 · · · sn = s. But s is an atom of S, which
implies n = 1. Thus, (s) = t1 and is an atom of T .
To prove the converse, suppose s 2 S such that (s) is an atom of T . Further suppose
that s = s1 s2 . Since
is a homomorphism, (s) = (s1 ) (s2 ). Now, (s) is an atom, so
either (s1 ) or (s2 ) must be a unit of T . Thus either s1 or s2 must be a unit of S, proving
that s is an atom of S.
To prove (2), suppose n 2 LT ( (s)). Then (s) = t1 · · · tn with t1 , . . . , tn atoms of T .
Thus there exist atoms s1 , . . . , sn of S and a permutation
each i 2 [1, n], (si ) = ui t
(i) vi ,
: [1, n] ! [1, n] such that for
with each ui and vi a unit of T and such that s1 · · · sn =
s. Therefore s can be written as the product of n atoms and n 2 LS (s). To prove the
converse, suppose n 2 LS (s). Then s = s1 · · · sn with s1 , . . . , sn atoms of S. Since
is a
homomorphism, (s) = (s1 ) · · · (sn ). By (1), since si is an atom of S for each i 2 [1, n],
each (si ) is an atom in T . Thus n 2 LT ( (s)) and LS (s) = LT ( (s)).
Like transfer homomorphisms, weak transfer homomorphisms are useful tools that relate
factorization properties of various semigroups. Below we provide two simple examples of
weak transfer homomorphisms and illustrate how they can be used to study factorizations.
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
16
Example 2.7. Consider the following mappings.
• For any semigroup S, the identity map i : S ! S is a weak transfer homomorphism.
Since the homomorphism i only maps S to itself, this is not particularly useful when
studying factorizations, but we use this fact in a later argument.
• By Remark 2.3, det : Mn (D) ! D• is a transfer homomorphism when D is a principal
ideal domain. In Theorem 2.8, we prove that this implies the determinant map is also
a weak transfer homomorphism. Thus, by Theorem 2.6, A 2 Mn (Z) is an atom if and
only if det(A) is prime. Furthermore, for any A 2 Mn (Z), LMn (Z) (A) = {m}, where m
is the number of prime factors of det(A), counting multiplicity. This shows Mn (Z) is
half-factorial.
As can be seen in Theorem 2.6, transfer homomorphisms and weak transfer homomorphisms preserve many factorization-theoretic properties. The next two theorems show that
these two concepts are synonymous in certain situations.
Theorem 2.8. Let S and T be an atomic semigroups with identities and let T be commutative. If the map
: S ! T is a transfer homomorphism, then
is also a weak transfer
homomorphism.
Proof. Let
: S ! T be a transfer homomorphism. Then T = (S)T ⇥ and
1
(T ⇥ ) = S ⇥ .
Furthermore, since T is commutative, T = T ⇥ (S)T ⇥ . Let s 2 S such that (s) = t1 · · · tn
with t1 , . . . , tn atoms of T . We show there exist atoms s1 , . . . , sn of S and the identity
permutation such that s1 · · · sn = s and for each i 2 [1, n], (si ) = ui t
(i) vi
= ui ti , where vi
is the identity of T and ui is a unit of T . We do so by induction on n.
If n = 1, (s) is an atom of T . Then by Theorem 2.2, since
is a transfer homomorphism,
s is an atom of S and the result holds. Now assume that whenever (s) can be written as a
product of n
1 atoms of T , we can find atoms s1 . . . , sn
1
of S such that s1 · · · sn
1
= s and
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
for each i 2 [1, n
17
1], (si ) = ui ti , where ui a unit of T . Now assume (s) = t1 · · · tn with
t1 , . . . , tn atoms of T . Since
is a transfer homomorphism, we can find non-units a and b of
S such that (a) = u(t1 t2 · · · tn 1 ) and (b) = vtn where u and v are units of T and ab = s.
Note that, by Theorem 1.10, ut1 is an atom of T since u is a unit and t1 is an atom. Then
(a) is the product of n
atoms s1 , . . . , sn
1
1 atoms and therefore, by our inductive hypothesis, we can find
of S such that s1 · · · sn
1
= a, (s1 ) = µ(ut1 ), and for each i 2 [2, n
1],
(si ) = ui ti , with µ and ui units of T . Let u1 = µu, un = v, and b = sn . Then for each
i 2 [1, n], (si ) = ui ti with ui a unit of T and s1 · · · sn = ab = s.
The converse of Theorem 2.8 is not generally true. In fact, a weak transfer homomorphism
may exist even when no transfer homomorphism exits, as shown in [DB14, Example 4.5].
However, Theorem 2.9 illustrates that we have a partial converse when both the domain and
codomain are commutative semigroups.
Theorem 2.9. Let S and T be a commutative atomic semigroups with identity. If the map
: S ! T is a weak transfer homomorphism, then
Proof. Let
and
1
is also a transfer homomorphism.
: S ! T be a weak transfer homomorphism. Then T = T ⇥ (S)T ⇥ = (S)T ⇥
(T ⇥ ) = S ⇥ .
Let s 2 S and suppose that (s) = xy for some non-units x and y in T . Since T is atomic,
we can write x = t1 · · · tm and y = tm+1 · · · tn , where ti is an atom of T for each i 2 [1, n].
Then (s) = t1 · · · tn . Since
of S and a permutation
(si ) = u1i t
(i) u2i ,
is a weak transfer homomorphism, there exist atoms s1 , . . . , sn
: [1, n] ! [1, n] such that s = s1 · · · sn and for each i 2 [1, n],
where u1i , u2i 2 T ⇥ . Since T is commutative, we may write (si ) = ui t
where ui = u1i u2i . Note that, by Lemma 1.8, ui 2 T ⇥ . Then (s
i 2 [1, n].
1 (i)
)=u
1 (i)
(i)
ti for each
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
Let a = s
1 (1)
···s
1 (m)
(a) = (s
(b) = (s
where vx = u
1 (1)
1 (1)
and b = s
) · · · (s
1 (m+1)
···u
1 (m)
1 (m+1)
1 (m)
) · · · (s
)=u
1 (n)
and vy = u
···s
1 (1)
)=u
1 (m+1)
and vy are units in T . Furthermore, ab = s
1 (n)
. Then
t1 · · · u
1 (m+1)
1 (m)
tm = vx x and
tm+1 · · · u
···u
1 (1)
18
1 (n)
···s
1 (n)
tn = vy y,
. Note that, by Lemma 1.8, vx
1 (n)
= s1 · · · sn = s. Thus,
is a
transfer homomorphism.
The following result is useful in the creation of new weak transfer homomorphisms and
will be used several times throughout the thesis.
Theorem 2.10. The composition of two weak transfer homomorphisms is a weak transfer
homomorphism.
: S ! T and ✓ :
Proof. Let S, T , and W be atomic semigroups with identities and let
T ! W be weak transfer homomorphisms. We show ✓
is a weak transfer homomorphism.
First we show that the composition of two homomorphisms is a homomorphism. Consider
(✓
)(ab), with a, b 2 S. Since
is a homomorphism, (✓
Furthermore, since ✓ is a homomorphism, ✓
Thus, ✓
(a) (b) = ✓
)(ab) = ✓
(a) ✓
(b) = (✓
(a) (b) .
)(a)(✓
)(b).
is a homomorphism.
Now we prove property (1) of Definition 2.5. Suppose s 2 (✓
)(s) 2 W ⇥ . That is, ✓( (s)) 2 W ⇥ , so (s) 2 ✓ 1 (W ⇥ ), hence s 2
✓ 1 (W ⇥ ) = T ⇥ , this means s 2
(✓
(ab) = ✓
1
) 1 (W ⇥ ). Then (✓
1
✓ 1 (W ⇥ ) . Since
(T ⇥ ). Since ✓ 1 (T ⇥ ) = S ⇥ , s 2 S ⇥ , and therefore
) 1 (W ⇥ ) ✓ S ⇥ . Now assume s 2 S ⇥ . Then since
1
(T ⇥ ) = S ⇥ , there exists t 2 T ⇥
such that (s) = t. Furthermore, since t 2 T ⇥ and ✓ 1 (W ⇥ ) = T ⇥ , there exists w 2 W ⇥
such that ✓(t) = w. Thus (✓
(✓
) 1 (W ⇥ ) = S ⇥ .
)(s) = ✓(t) = w 2 W ⇥ and s 2 (✓
) 1 (W ⇥ ). Hence,
CHAPTER 2. WEAK TRANSFER HOMOMORPHISMS
19
Now suppose w 2 W . Since ✓ is a weak transfer homomorphism, there exist t 2 T and
u1 , v1 2 W ⇥ such that w = u1 ✓(t)v1 . Furthermore, since
is a weak transfer homomorphism,
there exist s 2 S and u2 , v2 2 T ⇥ such that t = u2 (s)v2 . Thus
w = u1 ✓(u2 (s)v2 )v1 = u3 (✓
)(s)v3 ,
where, by Lemma 1.8, u3 = u1 ✓(u2 ) 2 W ⇥ and v3 = ✓(v2 )v1 2 W ⇥ . Thus we have proven
property (1) of Definition 2.5.
Now we prove property (2) of Definition 2.5. Let s 2 S. Suppose (✓
)(s) = ✓( (s)) =
w1 · · · wn such that wi is an atom of W for each i 2 [1, n]. Since ✓ is a weak transfer
homomorphism, there exist atoms t1 , . . . , tn of T and a permutation
such that (s) = t1 · · · tn and ✓(ti ) = ui w
i 2 [1, n]. Now, since
S and a permutation
1 (i)
1
: [1, n] ! [1, n]
vi , where ui and vi is a unit of W for each
is a weak transfer homomorphism, there exists atoms s1 , . . . , sn of
2
: [1, n] ! [1, n] such that s = s1 · · · sn and (si ) = µi t
2 (i)
⌘i , where
µi , ⌘i 2 T ⇥ for each i 2 [1, n].
Then
(✓
⌘i ) = ✓(µi )✓(t
2 (i)
where, by Lemma 1.8, ũi = ✓(µi )u
2 (i)
3
)(si ) = ✓(µi t
=
1
2.
2 (i)
Therefore, ✓
)✓(⌘i ) = ✓(µi )(u
2 (i)
w
2 W ⇥ and ṽi = v
1 ( 2 (i))
2 (i)
v
2 (i)
)✓(⌘i ) = ũi w
3 (i)
ṽi ,
✓(⌘i ) 2 W ⇥ . Furthermore,
is a weak transfer homomorphism.
Chapter 3
Direct Products of Semigroups
It was shown in [DB14] that there exists a weak transfer homomorphism from Tn (D) to
(D• )n when D is a principal ideal domain. Our goal is to extend this result to the semigroup
of upper triangular matrices over any Dedekind-finite cancellative ring. In order to do so, we
first need some results about factorization in products of semigroups. We first characterize
units and atoms in these product of semigroups.
Lemma 3.1. Let n 2 N and let!S1 , . . . , Sn be semigroups with identities 11 , . . . , 1n . Then
⇥
n
Y
u = (u1 , . . . , un ) 2 S ⇥ =
Si
if and only if ui 2 Si⇥ for each i 2 [1, n].
i=1
Proof. Suppose ui 2 Si⇥ for each i 2 [1, n]. Then for each i 2 [1, n] there exists ui
such that ui ui
1
1
2 Si⇥
= 1i = ui 1 ui . Let v = (u1 1 , . . . , un 1 ). Then uv = (u1 u1 1 , . . . , un un 1 ) =
(11 , . . . , 1n ) = (u1 1 u1 , . . . , un 1 un ) = vu. Thus, u 2 S ⇥ .
Now suppose u 2 S ⇥ . Then there exists v = (v1 , . . . , vn ) 2 S ⇥ such that uv = 1S = vu.
That is, for each i 2 [1, n], ui vi = 1i = vi ui . Thus, ui 2 Si⇥ for all i 2 [1, n].
Next we characterize atoms in products of semigroups. Note that while Lemma 3.1 holds
for all semigroups, we must restrict Lemma 3.2 to Dedekind-finite semigroups.
20
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
21
Lemma 3.2. Let n 2 N and let S1 , . . . , Sn be atomic Dedekind-finite semigroups with
n
Y
identities 11 , . . . , 1n . Then a = (a1 , . . . , an ) is an atom of S =
Si if an only if there exists
i=1
h 2 [1, n] with ah an atom of Sh and ai 2 Si⇥ for each i 2 [1, n]\{h}.
Proof. Assume ai 2 S ⇥ for each i 2 [1, n]. Then, by Lemma 3.1, a is a unit of S and
therefore is not an atom. Assume there exists h 2 [1, n] such that ah is a non-unit nonatom of Sh . Then ah = xy for some non-units x and y of Sh . Thus a = bc, where
b = (a1 , . . . , ah 1 , x, ah+1 , . . . , an ) and c = (11 , . . . , 1h 1 , y, 1h+1 , . . . , 1n ) with both x and y
in the hth position of their respective n-tuples. By Lemma 3.1, neither b nor c are units of S
since both have a non-unit in their hth position. Thus a is not an atom of S. Assume there
exist k, j 2 [1, n] with k 6= j and such that ak is an atom of Sk and aj is an atom of Sj . Then
a = bc, where b = (a1 , . . . , ak 1 , 1k , ak+1 , . . . , an ) and c = (11 , . . . , 1k 1 , ak , 1k+1 , . . . , 1n ),
with ak in the k th position. Since aj is not a unit, by Lemma 3.1, b is not a unit in S.
Similarly, since ak is not a unit, by Lemma 3.1, c is not a unit in S. In all of the cases above,
we have shown that a is not an atom of S.
Now suppose there exists h 2 [1, n] such that ah is an atom of Sh and ai 2 Si⇥ for all
i 2 [1, n]\{h}. Further suppose a = bc, with b = (b1 , . . . , bn ) and c = (c1 , . . . , cn ). Then for
each i 2 [1, n], ai = bi ci . Since for each j 2 [1, n]\{h}, aj 2 Sj⇥ and, by Lemma 1.8, since
Sj is Dedekind-finite, both bj and cj must be units of Sj . Also, since ah = bh ch and ah is an
atom of Sh , either bh or ch must be a unit of Sh . If bh is a unit, then bi 2 Si⇥ for all i 2 [1, n]
and thus, by Lemma 3.1, b is a unit in S. If ch is a unit, then ci 2 Si⇥ for all i 2 [1, n] and
thus, by Lemma 3.1, c is a unit in S. Thus, a is an atom in S.
Note that the statement “If a is an atom, then there exists h 2 [1, n] such that ah is
an atom of Sh and ai 2 Si⇥ for all i 2 [1, n]\{h}” is true regardless of the Dedekind-finite
condition. However, the converse of this statement is not generally true, as the next example
demonstrates.
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
22
Example 3.3. Consider Z• ⇥ T • , where T is the ring of linear transformations on an infinite
vector space as described in Example 1.9. Consider (3, i), where i is the identity transformation. Then 3 is an atom of Z• and i is a unit of T • . However, (3, i) = (3, R)(1, L), where,
by Lemma 3.1, neither (3, R) nor (1, L) are units of Z• ⇥ T • since neither R nor L are units
of T • . Thus, (3, i) is not an atom.
The following remark is used frequently to make inductive arguments in many of the
proofs of theorems involving large products of semigroups.
Remark 3.4. Let n 2 N and let S1 , . . . , Sn be semigroups. Then
!
n
Y1
Si ⇥ Sn .
n
Y
Si is isomorphic to
i=1
i=1
Since Dedekind-finiteness is such an important property for factorization, we prove the
following result.
Lemma 3.5. Let n 2 N and let S1 , . . . , Sn be atomic Dedekind-finite semigroups. Then
n
Y
Si is Dedekind-finite.
i=1
Proof. We prove the theorem by induction on n. Suppose S1 and S2 are Dedekind-finite
semigroups with identities 11 and 12 . Suppose u = (u1 , u2 ), v = (v1 , v2 ) 2 S1 ⇥ S2 such that
uv = (11 , 12 ). Then u1 v1 = 11 and u2 v2 = 12 . Thus, since both S1 and S2 are Dedekindfinite, v1 u1 = 11 and v2 u2 = 12 . Thus vu = (11 , 12 ) and S1 ⇥ S2 is Dedekind-finite.
Now assume the proposition holds for a product of n 1 semigroups. That is, if Si
n
Y1
is a Dedekind-finite semigroup for each i 2 [1, n 1],
Si is Dedekind-finite. Then by
i=1 !
n
Y1
both the inductive hypothesis and the n = 2 case,
Si ⇥ Sn is Dedekind-finite since
i=1
!
n
n
n
Y
Y1
Y
⇠
Sn is also Dedekind-finite. Hence, by Remark 3.4,
Si =
Si ⇥ Sn and
Si is
Dedekind-finite.
i=1
i=1
i=1
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
23
The following two results allow us to build weak transfer homomorphisms between direct
products of semigroups. In this way, we can construct weak transfer homomorphisms from
fairly complex products to much simpler ones, allowing us to prove several results, such as
Theorem 5.2.
Lemma 3.6. If S1 , S2 , T1 , and T2 are all atomic Dedekind-finite semigroups with identities
and
1
: S1 ! T1 and
2
: S2 ! T2 are both weak transfer homomorphisms, then ✓ :
S1 ⇥ S2 ! T1 ⇥ T2 defined by ✓((s1 , s2 )) = ( 1 (s1 ),
2 (s2 ))
is a weak transfer homomorphism.
Proof. Let u = (u1 , u2 ) 2 S1 ⇥ S2 and assume ✓(u) = ( 1 (u1 ),
Then, by Lemma 3.1,
1 (u1 )
2 T1⇥ and
2 (u2 )
2 T2⇥ . Since
1
2 (u2 ))
and
2
2 (T1 ⇥ T2 )⇥ .
are weak trans-
fer homomorphisms, u1 2 S1⇥ and u2 2 S2⇥ . Thus, by Lemma 3.1, u 2 (S1 ⇥ S2 )⇥ and
✓
1
(T1 ⇥ T2 )⇥ ✓ (S1 ⇥ S2 )⇥ . Now assume u = (u1 , u2 ) 2 (S1 ⇥ S2 )⇥ . Then, by Lemma
3.1, u1 2 S1⇥ and u2 2 S2⇥ . Again, since
1 (u1 )
2 T1⇥ and
2 (u2 )
Hence, (S1 ⇥ S2 )⇥ ✓ ✓
1
and
2
are weak transfer homomorphisms,
2 T2⇥ . Thus, by Lemma 3.1, ✓(u) = ( 1 (u1 ),
1
(T1 ⇥ T2 )⇥ and ✓
Now let t = (t1 , t2 ) 2 T1 ⇥ T2 . Since
s1 2 S1 such that t1 = u1
1 (s1 )v1 ,
1
1
2 (u2 ))
2 (T1 ⇥ T2 )⇥ .
(T1 ⇥ T2 )⇥ = (S1 ⇥ S2 )⇥ .
is a weak transfer homomorphism, there exists
where u1 , v1 2 T1⇥ . Similarly, since
homomorphism, there exists s2 2 S2 such that t2 = u2
2 (s2 )v2 ,
2
is a weak transfer
where u2 , v2 2 T2⇥ . Thus
t = u✓(s)v, with u = (u1 , u2 ), v = (v1 , v2 ), and s = (s1 , s2 ) 2 S1 ⇥ S2 . By Lemma 3.1,
u, v 2 (T1 ⇥ T2 )⇥ , and so T1 ⇥ T2 = (T1 ⇥ T2 )⇥ ✓(S1 ⇥ S2 ) (T1 ⇥ T2 )⇥ . Hence property (1) of
Definition 2.5 has been verified.
Let s = (s1 , s2 ) 2 S1 ⇥S2 be a non-unit and assume ✓(s) = t1 · · · tm such that ti = (ti1 , ti2 )
is an atom of T1 ⇥ T2 for all i 2 [1, m]. We show there exist atoms p1 , . . . , pm of S1 ⇥ S2
and a permutation
: [1, m] ! [1, m] such that s = p1 · · · pm and ✓(pi ) = ui t
(i) vi
for each
i 2 [1, m], with ui , vi 2 (T1 ⇥ T2 )⇥ . Consider the following three cases:
Case 1: If s2 2 S2⇥ , then since
2
is a weak transfer homomorphism,
2 (s2 )
2 T2⇥ . Then,
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
by Lemma 1.8, since T2 is Dedekind-finite and
2 (s2 )
=
m
Y
i=1
24
ti2 , ti2 2 T2⇥ for each i 2 [1, m].
By Lemma 3.2, since for each i 2 [1, m], ti2 2 T2⇥ , ti is an atom of T1 ⇥ T2 , and T1 and
T2 are both Dedekind-finite, ti1 is an atom of T1 for each i 2 [1, m]. Since
transfer homomorphism and
permutation
1 (s1 )
1
is a weak
= t11 · · · tm1 , there exist atoms s11 , . . . , sm1 of S1 and a
: [1, m] ! [1, m] such that s1 = s11 · · · sm1 and
1 (si1 )
= ui1 t
(i)1 vi1 ,
with
ui1 , vi1 2 T1⇥ .
Let p1 = (s11 , s2 ) and let pi = (si1 , 1) for i 2 [2, m]. Then
m
Y
pi = (s11 , s2 )
i=1
m
Y
(si1 , 1) = (s11 , s2 )
i=2
m
Y
=
si1 , s2
i=1
!
m
Y
si1 , 1
i=2
!
= (s1 , s2 ) = s.
Note that
✓(p1 ) =
1 (s11 ),
=
u11 t
=
u11 ,
= u1 t
2 (s2 )
(1)1 v11 ,
2 (s2 )
1
2 (s2 )t (1)2
t
(1)1 , t (1)2
(v11 , 1)
(1) v1 ,
1
2 (s2 )t (1)2
where, by Lemmas 1.8 and 3.1, u1 = u11 ,
and v1 = (v11 , 1) are both units of
T1 ⇥ T2 . Similarly, for all i 2 [2, m],
✓(pi ) =
1 (si1 ),
=
ui1 t
=
ui1 , t
= ui t
2 (1)
(i)1 vi1 , 1
1
(i)2
t
(i)1 , t (i)2
vi1 , 1
(i) vi ,
where, by Lemmas 1.8 and 3.1, ui = ui1 , t
1
(i)2
and vi = vi1 , 1 are both units of T1 ⇥ T2 .
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
25
Case 2: If s1 2 S1⇥ , we repeat the argument from Case 1 with the necessary alterations.
Case 3: If neither s1 nor s2 are units, then since both
are weak transfer
m
Y
homomorphisms, 1 (s1 ) and 2 (s2 ) are non-units. Thus, since 1 (s1 ) =
ti1 and 2 (s2 ) =
m
Y
i=1
1
and
2
i=1
ti2 , by Lemma 1.8, there exists k, j 2 [1, m] such that tk1 is a non-unit of T1 and tj2 is
a non-unit of T2 . Furthermore, by Lemma 3.2, since T1 and T2 are Dedekind-finite, each ti
has an atom in either the first or second position, with a unit in the other position. Let f
be the number of ti that have an atom in the first position. Further let K = {k1 , . . . , kf } =
{i 2 [1, n] : ti1 is an atom} with k1 < k2 < · · · < kf and let J = {j1 , . . . , jm
f}
= {i 2 [1, n] :
ti2 is an atom} with j1 < j2 < · · · < jm f .
m
Y
Consider 1 (s1 ) =
ti1 . Note that tk1 1 , . . . , tkf 1 are the atoms in this product, with
i=1
ti 2 T
⇥
if i 2
/ K. Thus, by Lemma 1.12, we may write
1 (s1 )
f
Y
=
rki 1 , where for each
i=1
i 2 [1, f ], rki 1 is an atom of T1 and rki 1 = uki 1 (tki 1 )vki 1 such that uki 1 , vki 1 2 T1⇥ . Since
1
is a weak transfer homomorphism, there exist atoms sk1 1 , . . . , skf 1 of T1 and a permutation
1
: K ! K such that s1 = sk1 1 · · · skf 1 and
where µki 1 , ⌘ki 1 2 T1⇥ .
Similarly,
2 (s2 ) =
1 (ski 1 )
= µki 1 (r
1 (ki )1
)⌘ki 1 for each i 2 [1, f ],
m f
Y
ti2 , and since tj1 2 , . . . , tjm
f2
are the atoms in this product with
i=1
ti2 2
T2⇥
i 2 [1, m
when i 2
/ J. Thus, by Lemma 1.12, we may write
2 (s2 )
rji 2 , where for each
i=1
f ], rji 2 is an atom of T2 and rji 2 = uji 2 (tji 2 )vji 2 such that uji 2 , vji 2 2 T2⇥ . Since
is a weak transfer homomorphism, there exists atoms sj1 2 , . . . sjm
2
=
f
Y
: J ! J such that s2 = sj1 2 · · · sjm
f2
and
2 (sji 2 )
= µji 2 (r
where µji 2 , ⌘ji 2 2 (T2 )⇥ . For i 2 [1, n], define
pi =
8
>
>
<(si1 , 1)
>
>
: 1, si2
if i 2 K
if i 2 J
.
f2
1 (ji )2
2
of T2 and a permutation
)⌘ji 2 for all i 2 [1, m
f ],
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
26
Note that sij is an atom of Sj , and thus, by Lemma 3.2, pi is an atom of S1 ⇥ S2 for each
i 2 [1, n].
Then
m
Y
pi =
i=1
f
Y
m f
Y
pki
i=1
i=1
f
Y
=
pj i
s ki 1 , 1
i=1
!
m f
Y
1,
sj i 2
i=1
= (s1 , 1)(1, s2 )
!
= (s1 , s1 ) = s.
Extend the permutations
i2
/ J. Then define
1
and
2
to [1, m] by fixing
: [1, m] ! [1, m] as the composition of
✓(pi ) =
1 (si1 ),
=
µi1 (r
=
µi1 u
= ui t
= i if i 2
/ K and
1 (i)
1
and
2.
2 (i)
= i if
If i 2 K,
2 (1)
1 (i)1
)⌘i1 , 1
1 (i)1
(t
1 (i)1
,t
1
(i)2
)v
1 (i)1
⌘i1 , 1
(i) vi ,
where, by Lemmas 1.8 and 3.1, ui = µi1 u
1 (i)1
and vi = v
1 (i)1
⌘i1 , 1 are both units
of T1 ⇥ T2 . Similarly, if i 2 J,
✓(pi ) =
1 (1),
=
1, µi2 (r
=
1, µi2 u
= ui t
2 (si2 )
2 (i)1
2 (i)2
(t
2 (i)2
)v
2 (i)2
⌘i2
(i) vi ,
where, by Lemmas 1.8 and 3.1, ui = ti11 , µi2 u
of T1 ⇥ T2 .
)⌘i2
2 (i)2
and vi = 1, v
2 (i)2
⌘i2 are both units
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
27
In each of these cases, we were able to express s as a product s = p1 · · · pm where, for
each i 2 [1, m], ✓(pi ) = ui t
(i) vi ,
with ui , vi 2 (T1 ⇥ T2 )⇥ . Thus, ✓ is a weak transfer
homomorphism.
We now extend the result in the lemma above to the product of arbitrarily many semigroups.
Theorem 3.7. Let n 2 N and for each i 2 [1, n], let Si and Ti be atomic Dedekind-finite
semigroups with identities and i : Si ! Ti be a weak transfer homomorphism. Then
n
n
Y
Y
✓ :
Si !
Ti defined by ✓ (s1 , . . . , sn ) =
1 (s1 ), . . . , n (sn ) is a weak transfer
i=1
i=1
homomorphism.
Proof. We prove the result by induction on n. The case n = 1 is trivial, and the case
n = 2 is proven by Lemma 3.6. Assume the result is true when S and T are products of
1 semigroups. That is, if for each i 2 [1, n
n
1], Si and Ti are both atomic semigroups
: Si ! Ti is a weak transfer homomorphism, then there exists a weak transfer
n
n
Y1
Y1
˜
homomorphism ✓ :
Si !
Ti defined by ✓˜ (s1 , . . . , sn 1 ) = 1 (s1 ), . . . , n 1 (sn 1 ) .
i=1
i=1
!
!
n
n
n
n
Y
Y1
Y
Y1
Si ⇠
Si ⇥ Sn and
Ti ⇠
Ti ⇥ Tn . Thus we can
Then by, Lemma 3.4,
=
=
and
i
express ✓ as ✓ :
n
Y1
Si
i=1
✓
⇣
(s1 , . . . , sn 1 ), sn
⌘
!
i=1
⇥ Sn !
i=1
n
Y1
i=1
Ti
!
i=1
⇥ Tn defined by
⇣
= ✓˜ (s1 , . . . , sn 1 ) ,
Then, by Lemma 3.6, since
n
i=1
n (sn )
⌘
=
⇣
1 (s1 ), . . . ,
n 1 (sn 1 )
,
n (sn )
⌘
.
is a weak transfer homomorphism by assumption and ✓˜ is
a weak transfer homomorphism by our inductive hypothesis, we know ✓ is a weak transfer
homomorphism.
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
28
Example 3.8. As stated in Example 2.7, the determinant map is a weak transfer homomorphism. Consider M = M2 (Z) ⇥ M3 (Z) ⇥ M4 (Z). Define ✓ : M ! (Z• )3 by ✓((A2 , A3 , A4 )) =
(det(A2 ), det(A3 ), det(A4 )), where A2 2 M2 (Z), A3 2 M3 (Z) and A4 2 M4 (Z). By Theorem
3.7, ✓ is a weak transfer homomorphism. Therefore we may study factorization of elements
in M by mapping them to (Z• )3 .
Theorem 3.7 allows us to map a product of semigroups to a simpler product of semigroups.
The next result shows that in certain products, we have an even nicer result.
Theorem 3.9. If D is a unique factorization domain, then
n
Y
(s1 , . . . , sn ) =
si is a transfer homomorphism.
: (D• )n ! D• defined by
i=1
• n
Proof. Let s = (s1 , . . . , sn ) 2 (D ) . Then (s) =
n
Y
si .
i=1
is a homomorphism. Let x, y 2 (D• )n with x = (x1 , . . . , xn ) and
n
n
n
Y
Y
Y
y = (y1 , . . . , yn ). Then, since D is commutative, (xy) =
xi yi =
xi
yi = (x) (y).
First we show that
i=1
To prove property (1) of Definition 2.5, note that if
n
Y
i=1
i=1
si is a unit in D, by Lemma 1.8,
i=1
since D is commutative and therefore Dedekind-finite, si is a unit for all i 2 [1, n]. Thus, by
Lemma 3.1, s is a unit in (D• )n . Hence
since for all x 2 D• ,
1
⇥
((D• )⇥ ) = ((D• )n ) . Furthermore,
(x, 1, . . . , 1) = x.
For property (2) of Definition 2.5, suppose (s) =
n
Y
is surjective
si = xy for some non-unit elements
i=1
x and y of D• . Since D is a unique factorization domain, we may write y = p1 · · · pm , where
pi is a prime in D for i 2 [1, m].
We induct on m. Suppose m = 1. Then y is prime. Thus there exists i 2 [1, n] such that
✓
◆
sk
y|si . Let k be the smallest such i. Let a = s1 , . . . , , . . . , sn and b = (1, . . . , y, . . . , 1),
y
(s)
where y is in the k th position. Clearly ab = s. Furthermore, (b) = y and (a) =
= x.
y
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
29
Now assume the result holds whenever y is a product of m
1 primes. That is, when
y = p1 · · · pm 1 , each pi a prime, and (s) = xy, there exists a and b in (D• )n such that
ab = s, (a) = ux and (b) = vy, where u and v are units in D• .
Suppose
(s) = xy with y = p1 · · · pm , with each pi prime.
Then
(s) = xy =
(xp1 )(p2 · · · pm ). Thus by our inductive hypothesis, there exist a and b in (D• )n such that
y
s = ab, (a) = xp1 and (b) =
. Furthermore, as shown in the case m = 1, there
p1
exist c and d in (D• )n such that a = cd, (c) = x and (d) = p1 . Thus, (c) = x,
✓ ◆
y
(db) = (d) (b) = p1
= y, and c(db) = (cd)b = ab = s.
p1
Example 3.10. Define
: (Z• )3 ! Z• by ((s1 , s2 , s3 )) = s1 s2 s3 . By Theorem 3.9, since
Z is a unique factorization domain, this is a transfer homomorphism, and therefore a weak
transfer homomorphism (Theorem 2.8). This means that since
((15, 42, 30)) =
18900 = 2 ·
3·
3·
7·
2·5·
5·
3,
we can write (15, 42, 30) as a product of length 8 with elements whose mappings are associate to those in the above product. Using the method described in the proofs of Theorems
3.9 and 2.8, we have that
(15, 42, 30) = (1, 1, 2)(1, 1, 3)(1, 3, 1)(1, 7, 1)(1, 2, 1)(1, 1, 5)( 5, 1, 1)( 3, 1, 1).
Theorem 3.9 is restricted to the case of unique factorization domains. The map
: Sn !
S is not generally a weak transfer homomorphism if S is any semigroup as shown in Example
3.12. However, we have the following result as consolation.
CHAPTER 3. DIRECT PRODUCTS OF SEMIGROUPS
30
Theorem 3.11. Let S be an atomic Dedekind-finite semigroup and let n 2 N. If ✓ : S n ! S
n
Y
is defined by ✓((a1 , . . . , an )) =
ai , then for any a 2 S n , LS n (a) ✓ LS ✓(a) .
i=1
Proof. Let a = (a1 , . . . , an ) 2 S n and m 2 LS n (a). Then there exist atoms b1 , . . . , bm of S n
such that a = b1 · · · bm where bi = (bi1 , . . . , bin ). Since bi is an atom for each i 2 [1, m],
by Lemma 3.2, there exists hi 2 [1, n] such that bihi is an atom of S and bij 2 S ⇥ for each
m
Y
j 2 [1, n]\{hi }. Furthermore, for each i 2 [1, n], ai =
bji .
j=1
Then ✓(a) = a1 · · · an = (b11 · · · bm1 ) · · · (b1n · · · bmn ). This is the product of exactly m
atoms, namely bihi , for each i 2 [1, m], with the remaining elements in the product units.
That is, bij 2 S ⇥ if (i, j) 2
/ {(i, hi ) : i 2 [1, m]}. Thus, by Lemma 1.12, ✓(a) can be written
as a product of m atoms, and m 2 LS ✓(a) .
The following example illustrates Theorem 3.11. It also provides a case where the converse
of Theorem 3.11 is false in the case of a semigroup that is not a unique factorization domain.
Example 3.12. Let P = Z[x3 , x4 , x5 ]• and consider a = (x4 , x6 ) 2 P 2 . The only way to
factor a as a product of atoms, up to associates and up to permutations, is as (x4 , x6 ) =
(x4 , 1)(1, x3 )2 . Thus, LP 2 (a) = {3}. If we define ✓ as in Theorem 3.11, we have ✓(a) =
x10 = (x3 )2 x4 = (x5 )2 . Thus, LP (✓(a)) = {2, 3}. As can be seen, LP 2 (a) ⇢ LP (✓(a)), but
LP (✓(a)) 6⇢ LP 2 (a).
We give one more result and corollary that provide useful facts about lengths sets of
elements in product semigroups.
Remark 3.13. Let n 2 N and for each i 2([1, n], let Si be an atomic
semigroup. Then if
)
n
n
Y
X
a = (a1 , . . . , an ) 2
Si = S, then LS (a) =
mi : mi 2 LSi (ai ) .
i=1
i=1
Corollary 3.14. Let n 2 N and for each i 2 [1, n], let Si be an atomic semigroup. Then
n
Y
Si is half-factorial if and only if for each i 2 [1, n], Si is half-factorial.
i=1
Proof. This follows directly from Remark 3.13.
Chapter 4
Generalized Block Triangular
Matrices
We now turn our attention to the factorization of matrices. To do so, we introduce the
semigroup of focus. While studying factorization of matrices with non-commutative entries,
as well as block matrices, it becomes useful to consider the following semigroup.
Definition 4.1. Generalized Block Triangular Matrices
k
X
Let R be a ring, k, m1 , . . . , mk 2 N, n =
mi , and for each i 2 [1, k], let Si be a
i=1
non-empty subsemigroup of Mmi (R). Define T (S1 , . . . , Sk ) to be the set of all n ⇥ n matrices
of the form
2
6 A1 A12
6
6021 A2
6
A=6 .
..
6 ..
.
6
4
0k1 0k2
3
. . . A1k 7
7
. . . A2k 7
7
,
.. 7
..
7
.
. 7
5
. . . Ak
where Ai 2 Si for all i 2 [1, k], Aij 2 Mmi ,mj (R), and 0ij is the mi ⇥ mj zero matrix.
That is, T (S1 , . . . , Sk ) is the set of all block triangular matrices with diagonal entries in the
semigroups S1 , . . . , Sk .
31
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
32
For ease of notation, we will often write 0ij as 0 and Ii , the identity of Mi , as I, with the
understanding that each must be of the proper size.
We now show that T (S1 , . . . , Sk ) is indeed a semigroup under usual matrix multiplication
in Mn (R).
Theorem 4.2. Let T (S1 , . . . , Sk ) be defined as in Definition 4.1. Then T (S1 , . . . , Sk ) is a
subsemigroup of Mn (R).
Proof. Clearly T (S1 , . . . , Sk ) is a non-empty subset of Mn (R). We show T (S1 , . . . , Sk ) is a
semigroup closed
2
6A. 1
.
Let A = 6
4 .
0
under usual
3 matrix
2 multiplication.
3
. . . A1k 7
6B. 1 .. . . B.1k 7
.. 7
...
6 ..
..
.. 7 2 T (S1 , . . . , Sk ). Note that the j th entry
,
B
=
. 5
4
5
. . . Ak
0 . . . Bk
of the ith row of A, Aij , is of size i ⇥ j. Furthermore, the j th entry of the ith column
of B, B3ji ,
2
6C.1 .. . . C.1k 7
.
. . .. 7 .
is of size j ⇥ i. Therefore the matrix product AB is defined. Write AB = 6
4 .
5
0 . . . Ck
We show AB 2 T (S1 , . . . , Sk ) by showing Ci 2 Si for each i 2 [1, k]. By usual matrix
multiplication,
Ci = 0B1i + · · · + 0B(i
1)i
+ Ai Bi + Ai(i+1) 0 + · · · + Aik 0 = Ai Bi .
Since Ai 2 Si , Bi 2 Si , and Si is a semigroup, Ci 2 Si for all i 2 [1, k]. Therefore
AB 2 T (S1 , . . . , Sk ) and T (S1 , . . . , Sk ) is closed under multiplication. Thus T (S1 , . . . , Sk ) is
a subsemigroup of Mn (R).
Definition 4.1 is very general. Each of the semigroups along the diagonal need not be
of the same size or even type (upper triangular, lower triangular, diagonal, etc.). As such,
many other matrix semigroups can be written as a generalized block triangular matrix. The
following example shows a few such semigroups.
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
33
Example 4.3. Each of the following semigroups can be represented as generalized block
triangular matrices.
1. Any ring R can be considered as a ring of 1 ⇥ 1 matrices. Consequently, Tn (R) ⇠
=
T (R, . . . , R).
2. If R is any ring and k, n 2 N, then Tk (Mn (R)) ⇠
= T (Mn (R), . . . , Mn (R)).
3. Let P = Z[x3 , x4 , x5 ]• . If U T2 (P) is the semigroup of 2 ⇥ 2 upper triangular matrices,
Mn (Z) is the semigroup of 2 ⇥ 2 integer matrices, and LT3 (P) is the semigroup of 3 ⇥ 3
⇣
⌘
lower triangular matrices, then T = T U T2 (P), M2 (Z), LT3 (P) is a generalized block
triangular matrix semigroup. We show an example of an element in T :
2⇥
6
6
6
6
4
x15 x9
0 x4
⇤ ⇥ x2
x3
x9 x7
⇤
[ 87 10
8 ]
0
0
0
⇥ x3
⇤3
7
7
⇤
3
7.
x
3
7
x
i5
0
x4 1
x4 x6 x10
⇥ x5
x9
h x10
x5
x4
x12
1
0
1 0
x9 x15
Since T (S1 , . . . , Sk ) is a subsemigroup of the semigroup of all n ⇥ n matrices, it is natural
to consider embedding T (S1 , . . . , Sk ) within a larger generalized block triangular matrix. We
do so in Lemma 4.4 which will be used frequently as we employ inductive arguments to study
T (S1 , . . . , Sk ).
Lemma 4.4. If T (S1 , . . . , Sk ) is defined as in Definition 4.1, then T (S1 , . . . , Sk ) is isomorphic
to T T (S1 , . . . , Sk 1 ), Sk .
Proof. Define ✓ : T (S1 , . . . , Sk ) ! T T (S1 , . . . , Sk 1 ), Sk by ✓(A) =
2
3
2
6A. 1 .. . . A.1k 7 0 6A. 1 .. . . A1(k.
6 ..
..
.. 7, A = 6 ..
..
..
4
5
4
0 . . . Ak
0 . . . Ak
3
3
6 A1k 7
7, and A00 = 6 ... 7.
5
4
5
A(k 1)k
1) 7
1
2
"
0
A
0
00
#
A
, where A =
Ak
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
34
Clearly ✓ is an bijection and the identity matrix is mapped to the identity matrix. We
show ✓ is a semigroup homomorphism. Let B, A 2 T (S1 , . . . , Sk ). Then
2
6 A1 B 1
6
6
6
6
"
#"
# 6 0
6
A1 ... A1k
B1 ... B1k
6 .
.. . . ..
AB = ... . . . ...
=
6 .
. . .
6 .
0 ... Ak
0 ... Bk
6
6
6 0
6
6
4
2
X
and so ✓(AB) =
C
0
2
6 A1 B 1
6
6
6
6
6
C0 = 6 0
6
6 .
6 ..
6
4
0
0
00
A2 B 2
i=1
k 1
X
...
A1i Bi(k
A2i Bi(k
i=2
..
.
..
0
...
0
...
.
Ak
1)
1)
..
.
1 Bk 1
0
k
X
C
, where
Ak Bk
2
X
k 1
X
A1i Bi2 . . .
i=1
A1i Bi(k
i=1
A2 B2
k 1
X
...
..
.
...
0
...
i=2
A2i Bi(k
..
.
Ak 1 B k
1
2
3
(k 1)i
i=k 1
2
2
6A
✓(A)✓(B) = 4
0
0
3
3
6 B1k 7
7, and B 00 = 6 ... 7,
5
4
5
B(k 1)k
1) 7
1
2
32
3
2
3
k
X
6
A1i Bik
6
7
6 i=1
7
6 X
k
7
6
7
6
A2i Bik
6
1) 7
7 and C 00 = 6 i=2
7
6
..
7
6
.
7
6
7
6 k
6X
5
4
A
B
1) 7
3
7
7
7
7
A2i Bik 7
7
7
i=2
7
..
7.
.
7
7
k
7
X
A(k 1)i Bik 7
7
7
i=k 1
5
A1i Bik
i=1
k
X
Ak Bk
#
6B. 1 .. . . B1(k.
.
..
..
Then, if B = 6
4 .
0 . . . Bk
0
A1i Bi2 . . .
i=1
0
"
k 1
X
3
A00 7 6B 0 B 00 7 6A0 B 0 A0 B 00 + A00 Bk 7
54
5=4
5.
Ak
0 Bk
0
Ak Bk
ik
7
7
7
7
7
7
7
7.
7
7
7
7
7
5
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
35
Clearly, A0 B 0 = C 0 . Furthermore,
2
k 1
X
3
2
3
2
k
X
3
6
A1i Bik 7 6 A1k Bk 7 6
A1i Bik 7
6
7 6
7 6
7
6 i=1
7 6
7 6 i=1
7
6 X
7 6
7 6 X
7
k 1
k
6
7 6
7 6
7
6
7
6
7
6
7
A
B
A
B
A
B
2k
k
2i
ik
2i
ik
6
7 6
7 6
7
0 00
00
A B + A Bk = 6 i=2
7+6
7 = 6 i=2
7 = C 00 .
6
7 6
7 6
7
..
..
..
6
7 6
7 6
7
.
.
.
6
7 6
7 6
7
6k 1
7 6
7 6 k
7
6X
7 6
7 6X
7
4
A(k 1)i Bik 5 4A(k 1)k Bk 5 4
A(k 1)i Bik 5
i=k 1
i=k 1
Thus ✓(AB) = ✓(A)✓(B) and ✓ is an isomorphism.
We now characterize the units and atoms of a semigroup of generalized block triangular
matrices. In Lemmas 4.5 and 4.6, we consider the 2 ⇥ 2 case. We then use an inductive
argument to give general results in Theorems 4.7 and 4.8.
Lemma 4.5. Let R be a ring with unity, n, m 2 N, S1 a subsemigroup of Mm (R), and S2 a
"
#
A B
subsemigroup of Mn (R). Let M =
2 T (S1 , S2 ). The matrix M is a unit of T (S1 , S2 )
0 C
if and only if A is a unit of S1 and C is a unit of S2 .
Proof. Assume M is a unit of T (S1 , S2 ). Then there exists N 2 T (S1 , S2 ) such that M N =
"
# "
#"
# "
# "
#"
#
D E
A B D E
I
0
D E A B
Im+n = N M . That is, if N =
,
= m
=
.
0 F
0 C
0 F
0 In
0 F
0 C
Since In = 0E + CF = CF and In = 0B + F C = F C, C is a unit in S2 . Furthermore, since
Im = AD + B0 = AD and Im = DA + E0 = DA, A is a unit in S1 .
2
6A
Now assume both A and C are units in their respective semigroups. Let N = 4
0
1
1
A BC
C
1
1
3
7
5.
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
36
We show M N = Im+n = N M . First
2
32
1
1
1
3
A BC 7
6A B 7 6A
MN = 4
54
5
0 C
0
C 1
2
1
1
1
6AA + B0 A( A BC ) + BC
= 4
0A 1 + C0 0( A 1 BC 1 ) + CC
2
3
6I m 0 7
= 4
5.
0 In
1
1
3
7
5
Also
2
1
1
1
32
3
A BC 7 6A B 7
6A
NM = 4
54
5
0
C 1
0 C
2
1
1
1
1
1
6A A + ( A BC )0 A B + ( A BC
= 4
0A + C 1 0
0B + C 1 C
2
3
6Im 0 7
= 4
5.
0 In
1
3
)C 7
5
Thus, M is a unit of T (S1 , S2 ).
With units in the 2 ⇥ 2 case categorized, we now consider atoms. The proof of Lemma
4.6 is similar to that found in Lemma 3.2.
Lemma 4.6. Let R be a ring with unity, S1 a Dedekind-finite subsemigroup of Mm (R), and
"
#
A B
S2 a Dedekind-finite subsemigroup of Mn (R). Let M 2 T (S1 , S2 ) with M =
. The
0 C
matrix M is an atom of T (S1 , S2 ) if and only if exactly one of A and C is an atom in its
respective matrix semigroup while the other is a unit.
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
37
"
#
A B
Proof. Let M =
2 T (S1 , S2 ). Consider the following cases. If both A and C are
0 C
units in their respective semigroups, then by Theorem
4.5,
"
# " M is#a unit and thus not an
atom. If both A and C are non-units, then M =
"
Im B
0 C
A 0
, where, by Lemma 4.5,
0 In
#
"
#
Im B
A 0
both
and
are non-units since they contain a non-unit along their diagonal.
0 C
0 In
Thus M can be written as a product of two non-units and is therefore, by Lemma 1.8, not
an atom. If A "is a non-unit
then A = A1 A2 for some
A
# " non-atom,
#
" non-units
#
"1 and A
# 2 of
A1 B A2 0
A1 B
A2 0
S1 . Then M =
, where, by Lemma 4.5, both
and
are
0 C
0 In
0 C
0 In
non-units, and thus M is not an atom. If C is"a non-unit
then C = C1 C2 for
# " non-atom,
#
some non-units C1 and C2 of S2 . Then M =
"
#
"
#
Im 0
0 C1
A B
, where, by Lemma 4.5,
0 C2
Im 0
A B
and
are non-units, and thus M is not an atom. Therefore if M is
0 C1
0 C2
an atom, exactly one of A and C is an atom in its respective matrix group, while the other
both
is a unit.
⇥
Now we show the converse. Assume A is2 an atom
3 of S1 and C 2 S2 . Suppose M =
"
#
"
#
6G H 7
A B
D E
= KL with K =
,L = 4
5 2 T (S1 , S2 ). Then C = F J, and,
0 C
0 F
0 J
by Lemma 1.8, since S2 is Dedekind-finite and C is a unit, both F and J must be units.
Furthermore, A = DG, and since A is an atom, by Lemma 1.10,either D or G must be a
unit. If D is a unit, then since F is also a unit, by Lemma 4.5, K is a unit of T (S1 , S2 ).
Similarly, if G is a unit, then since J is also a unit, by Lemma 4.5, L is a unit of T (S1 , S2 ).
In both cases, one of K or L is a unit, and thus M is an atom of T (S1 , S2 ).
Alternatively, assume C is an atom of S2 and A 2 S1⇥ . Suppose M =
"
#
"
#
"
A B
0 C
#
= KL
D E
G H
,L =
2 T (S1 , S2 ). Then A = DG, and, by Lemma 1.8, since S1
0 F
0 J
is Dedekind-finite and A is a unit, both D and G must be units. Furthermore, C = F J, and
with K =
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
38
since C is an atom, by Lemma 1.10, either F or J must be a unit. If F is a unit, since D is
also a unit, by Lemma 4.5, K is a unit of T (S1 , S2 ). Similarly, if J is a unit, since G is also
a unit, by Lemma 4.5, L is a unit of T (S1 , S2 ). In both cases, one of K or L is a unit, and
thus M is an atom of T (S1 , S2 ).
We now generalize Lemma 4.5 to generalized block triangular matrices of arbitrary size.
2
3
6A. 1 .. . . A.1k 7
.
..
.. 7 2
Theorem 4.7. Let T (S1 , . . . , Sk ) be defined as in Definition 4.1. Let A = 6
4 .
5
0 . . . Ak
T (S1 , . . . , Sk ). The matrix A is a unit in T (S1 , . . . , Sk ) if and only if Ai is a unit in Si for
each i 2 [1, k].
Proof. We prove the result by induction on k. The case k = 1 is trivial and the case k = 2 is
proven by Lemma 4.5. Assume the result holds for size k
1. That is, M 2 T (S1 , . . . , Sk 1 )⇥
if and only if each of it’s diagonal matrices are units of their respective semigroup.
By
"
#
0
00
A A
Lemma 4.4, T (S1 , . . . , Sk ) ⇠
,
= T T (S1 , . . . , Sk 1 ), Sk , and so we write A as A =
0 Ak
2
3
2
3
6A. 1 .. . . A1(k. 1) 7
6 A1k 7
..
..
.. 7 2 T (S1 , . . . , Sk 1 ) and A00 = 6 ... 7.
where A0 = 6
4
5
4
5
0 . . . Ak 1
A(k 1)k
Assume A is a unit. Then by Lemma 4.5, A0 is a unit of T (S1 , . . . , Sk 1 ) and Ak is a unit
of Sk . Then by the inductive hypothesis, since A0 is a unit of T (S1 , . . . , Sk 1 ), Ai is a unit
of Si for each i 2 [1, k
1]. Thus, Ai is a unit of Si for each i 2 [1, k].
Now assume Ai is a unit of Si for each i 2 [1, k]. Then by the induction hypothesis, A0
is a unit of T (S1 , . . . , Sk 1 ). Thus, by Lemma 4.5, A is a unit of T T (S1 , . . . , Sk 1 ), Sk ⇠
=
T (S1 , . . . , Sk ).
CHAPTER 4. GENERALIZED BLOCK TRIANGULAR MATRICES
39
We now generalize Lemma 4.6 to generalized block triangular matrices of arbitrary size.
Theorem 4.8. Let T (S1 , . . . , Sk ) be2defined as in 3Definition 4.1 such that Si is Dedekind6A. 1 .. . . A.1k 7
.
..
.. 7 2 T (S1 , . . . , Sk ). The matrix A is an
finite for each i 2 [1, k]. Let A = 6
4 .
5
0 . . . Ak
atom of T (S1 , . . . , Sk ) if and only if there exists h 2 [1, k] such that Ah is an atom of Sh and
Ai 2 Si⇥ for each i 2 [1, k]\{h}.
Proof. We prove the result by induction on k. The case k = 1 is trivial2and the case k = 32 is
proven by Lemma 4.6. Assume the result holds for size k
6A. 1 .. . . A1(k.
.
..
..
1. That is, 6
4 .
0 . . . Ak
T (S1 , . . . , Sk 1 ) is an atom if and only if there exists h 2 [1, k
1) 7
1
72
5
1] such that Ah is an
1]\{h}. By Lemma 4.4, T2(S1 , . . . , Sk ) ⇠
= 3
"
#
0
00
6A. 1 .. . . A1(k. 1) 7
A A
..
..
.. 7 2
T T (S1 , . . . , Sk 1 ), Sk , and so we may write A as A =
, where A0 = 6
4
5
0 Ak
0 . . . Ak 1
2
3
6 A.1k 7
00
. 7
T (S1 , . . . , Sk 1 ) and A = 6
4 . 5.
A(k 1)k
atom of Sh and Ai 2 Si⇥ for each i 2 [1, k
Assume A is an atom. Then by Lemma 4.6, either A0 is an atom of T (S1 , . . . , Sk 1 ) and
Ak 2 Sk⇥ or A0 2 T (S1 , . . . , Sk 1 )⇥ and Ak is an atom of Sk . If A0 is an atom, then by the
inductive hypothesis, there exists h 2 [1, k 1] such that Ah is an atom of Sh and Ai 2 Si⇥ for
each i 2 [1, k
1]\{h}. Then since Ak is a unit, Ai 2 Si⇥ for each i 2 [1, k]\{h}. If Ak is an
atom then A0 2 T (S1 , . . . , Sk 1 )⇥ and thus, by Theorem 4.7, Ai 2 Si⇥ for each i 2 [1, k
1].
Now assume there exists h 2 [1, k] such that Ah is an atom of Sh and Ai 2 Si⇥ for each i 2
[1, k]\{h}. If h = k, then Ak is an atom and Ai 2 Si⇥ for each i 2 [1, k 1]. Thus, by Theorem
4.7, A0 2 T (S1 , . . . , Sk 1 )⇥ . Hence, by Lemma 4.6, A is an atom of T (T (S1 , . . . , Sk 1 ), Sk ) ⇠
=
T (S1 , . . . , Sk ). If h 6= k, h 2 [1, k
1] and Ak 2 Sk⇥ . Thus, by our inductive hypothesis, A0 is
an atom. Hence, by Lemma 4.6, A is an atom of T (T (S1 , . . . , Sk 1 ), Sk ) ⇠
= T (S1 , . . . , Sk ).
Chapter 5
Factoring Block Triangular Matrices
With units and atoms of T (S1 , . . . , Sk ) characterized, we now study factorizations of generalized block triangular matrices. We do so by constructing a weak transfer homomorphism
to a product of semigroups. Again, we first consider the 2 ⇥ 2 case (Lemma 5.1) and then
generalize to the arbitrary case (Theorem 5.2). The construction in Lemma 5.1 is similar to
the construction provided in Lemma 3.6.
Lemma 5.1. Let R be a Dedekind-finite ring with unity, S1 an atomic Dedekind-finite
subsemigroup of Mm (R), and S2 an atomic Dedekind-finite subsemigroup of Mn (R). Define
"
#!
A B
✓ : T (S1 , S2 ) ! S1 ⇥ S2 by ✓
= (A, C). The function ✓ is a weak transfer
0 C
homomorphism.
"
#
A 0
Proof. If (A, C) 2 S1 ⇥ S2 , then M =
2 T (S1 , S2 ) and ✓(M ) = (A, C). Thus ✓ is
0 C
surjective.
"
#
A B
2 T (S1 , S2 )⇥ , then, by Lemma 4.5, A 2 S1⇥ and C 2 S2⇥ . Thus,
0 C
"
#
A B
⇥
by Lemma 3.1, ✓(M ) = (A, B) 2 (S1 ⇥ S2 ) . Also, if M =
2 T (S1 , S2 ) such that
0 C
Now, if M =
40
CHAPTER 5. FACTORING BLOCK TRIANGULAR MATRICES
41
✓(M ) = (A, B) 2 (S1 ⇥ S2 )⇥ , then by Lemma 3.1, A 2 S1⇥ and B 2 S2⇥ . Thus, by Lemma
4.5, M 2 T (S1 , S2 )⇥ . Hence we have proven property (1) of Definition 2.5.
"
#
A B
Let M =
2 T (S1 , S2 ) and suppose ✓(M ) = (A, C) = t1 · · · tm , where each
0 C
ti = (Di , Ei ) is an atom of S1 ⇥ S2 . Note that since each ti is an atom, by Lemma 4.6,
exactly one of Di and Ei must be an atom with the other a unit. We show there exist atoms
: [1, m] ! [1, m] such that M = P1 · · · Pm , and
P1 , . . . , Pm of T (S1 , S2 ) and a permutation
with ui , vi 2 (S1 ⇥ S2 )⇥ . We consider three cases.
m
Y
Case 1: If C is a unit, by Lemma 1.8, since S2 is Dedekind-finite and C =
Ei , Ei is a
for each i 2 [1, m], ✓(Pi ) = ui t
(i) vi ,
i=1
unit for each i 2 [1, m]. Thus, by Lemma 3.2, since each ti is an atom, each Di must be an
m
Y
atom of S1 . Also note that A =
Di .
Define P1 =
"
m
Y
i=1
#
i=1
"
#
D1 B
Di 0
and Pi =
for each i 2 [2, m]. Then
0 C
0 I
Pi
32 m
3
30 2
31 2
Y
m
D1 B 7 6 Di 0 7
6D1 B 7 BY 6Di 07C 6
7
7 6 i=2
= 4
5@ 4
5A = 6
4
54
5
0 C
0 I
i=2
0 C
0
I
2m
3 2
3
Y
6 Di B 7 6A B 7
7=4
= 6
5 = M.
4 i=1
5
0 C
0
C
2
Furthermore, ✓(P1 ) = (D1 , C) = (I, CE1 1 )t1 (I, I), and for each i 2 [2, m], ✓(Pi ) = (Di , I) =
(I, Ei 1 )ti (I, I). By Lemmas 1.8 and 3.1, (I, CE1 1 ), (I, I), and (I, Ei 1 ) are all units of
S1 ⇥ S2 .
Case 2: If A is a unit, by Lemma 1.8, since S1 is Dedekind-finite and A =
m
Y
i=1
Di , Di is a
unit for each i 2 [1, m]. Thus, by Lemma 3.2, since each ti is an atom, each Ei must be an
m
Y
atom of S2 . Also, C =
Ei .
i=1
CHAPTER 5. FACTORING BLOCK TRIANGULAR MATRICES
"
#
I 0
Define Pi =
for each i 2 [1, m
0 Ei
0
m
Y
2
"
#
A B
1] and Pm =
. Then
0 Em
31 2
3
2
I
m 1
B Y 6I 0 7C 6A B 7 6
Pi = @
4
5A 4
5=6
4
0
0 Ei
0 Em
i=1
i=1
2
3
2
42
3
B 7
6A
6A B 7
m
Y 7
= 6
5 = M.
4
5=4
0
Ei
0 C
0
m
Y1
i=1
Ei
32
76
76
54
A
B
0 Em
3
7
7
5
i=1
Furthermore, ✓(Pm ) = (A, Em ) = (ADm1 , I)tm (I, I) and for each i 2 [1, m
1], ✓(Pi ) =
(I, Ei ) = (Di 1 , I)ti (I, I). By Lemmas 1.8 and 3.1, (ADm1 , I), (I, I), and (Di 1 , I) are all
units of S1 ⇥ S2 .
Case 3: Suppose neither A nor B is a unit. Then, by Lemma 1.8, since A =
there exists k 2 [1, m] such that Dk is a non-unit of S1 . Similarly, since B =
m
Y
m
Y
Di ,
i=1
Ei , there
i=1
exists j 2 [1, m] such that Ej is a non-unit of S2 . Then, by Lemma 3.1, since S1 and S2 are
Dedekind-finite and ti is an atom of S1 ⇥ S2 , for each i 2 [1, m], exactly one of Di and Ei
is an atom of their respective group while the other is a unit. Let f represent the number
of indicies i such that Di is an atom. Let K = {k1 , . . . , kf } = {i 2 [1, n] : Di is an atom}
with k1 < k2 < · · · < kf . Also let J = {j1 , · · · , jm
j1 < j2 < · · · < jm
Note that A =
f.
m
Y
f}
= {i 2 [1, n] : Ei is an atom} with
Di , where Dk1 , . . . , Dkf are precisely the atoms of this product and
i=1
Di 2
S1⇥
for all i 2
/ K. Then, by Lemma 1.12, we may write A =
f
Y
Ai , where for each
i=1
i 2 [1, f ], Ai = Ui Dki Vi with Ui and Vi both units of S1 . Similarly, note that C =
where Ej1 , . . . , Ejm
f
m
Y
i=1
are precisely the atoms of this product and Ei 2 S2⇥ for all i 2
/ J.
Ei ,
CHAPTER 5. FACTORING BLOCK TRIANGULAR MATRICES
43
m f
Then, by Lemma 1.12, we may write C =
Y
i=1
Ri Eji Qi with Ri and Qi units of S2 .
82
>
>
>
>
6I
>
>
4
>
>
>
0
>
>
>
>
2
>
>
>
<
6I
For each i 2 [1, m] define Pi = 4
>
>
0
>
>
>
2
>
>
>
>
>
>
6Ai
>
>
4
>
>
>
:
Then
m
Y
0
m f 1
2
31 2
Ci , where for each i 2 [1, m
3
07
5
Ci
3
B 7
5
Cm
f
(m f )
0
30
if i < m
f
if i = m
f
3
07
5 if i > m
I
m
Y
2
B 7B
B Y 6I 0 7C 6I
6Ai (m
Pi = @
4
5A 4
5@
4
0 Ci
0 Cm f
0
i=1
i=1
i=m f +1
0
2
31 2
30 2
31
m f 1
f
B 7 BY
B Y 6I 0 7C 6I
6Ai 07C
=@
4
5A 4
5@ 4
5A
0 Ci
0 Cm f
0 I
i=1
i=1
2
32
3
3 2Y
f
I
0
6
7 6I
B 7 6 Ai 0 7
6
7
m
f
=4
4
56
Y1 7
5
4 i=1
5
0
Ci
0 Cm f
0
I
i=1
2
32 f
3
Y
I
B
6
7 6 Ai 07
7 6 i=1
7
m f
=6
Y
4
54
5
0
Ci
0
I
i=1
2
32
3
6I B 7 6A 0 7
=4
54
5
0 C
0 I
2
3
6A B 7
=4
5 = M.
0 C
f.
f)
31
0 7C
5A
I
f ], Ci =
CHAPTER 5. FACTORING BLOCK TRIANGULAR MATRICES
Define the permutation
If i  m
: [1, m] ! [1, m] by (i) =
f,
8
>
>
<j
i
>
>
:k(i
(m f ))
44
if i  m
f
if i > m
f
.
✓(Pi ) = (I, Ci )
= (I, Ri Eji Qi )
= (D
where, by Lemma 3.1, (D
Similarly, if i > m
1
(i) , Ri )
1
(i) , Ri )t (i) (I, Qi )
and (I, Qi ) are units of S1 ⇥ S2 .
f,
✓(Pi ) = (Ai
(m f ) , I)
= (Ui
(m f ) Dki
= (Ui
1
(m f ) , E (i) )t (i) (Vi (m f ) , I)
where, by Lemma 3.1, (Ui
1
(m f ) , E (i) )
and (Vi
(m f )
Vi
(m f ) , I)
(m f ) , I)
are units of S1 ⇥ S2 .
In each of these cases, we have been able to express M as a product M = P1 · · · Pm where
for each i 2 [1, m], ✓(Pi ) = ui t
(i) vi ,
with ui , vi 2 (S1 ⇥ S2 )⇥ . Thus, ✓ is a weak transfer
homomorphism.
We now extend Lemma 5.1 to generalized block triangular matrices of arbitrary size.
This is the main result of the thesis.
Theorem 5.2. Let R be a Dedekind-finite ring with unity, k, m1 , . . . , mk 2 N and, for
each i 2 [1, k], let Si be an0atomic
Dedekind-finite
subsemigroup of Mmi (R). Define ✓ :
2
31
k
Y
B6A. 1 .. . . A.1k 7C
6 .
..
.. 7C = (A1 , . . . , Ak ). The function ✓ is a weak
T (S1 , . . . , Sk ) !
Si by ✓ B
@4 .
5A
i=1
0 . . . Ak
transfer homomorphism.
CHAPTER 5. FACTORING BLOCK TRIANGULAR MATRICES
45
Proof. We prove the result by induction on k. The case k = 1 is trivial and the case k = 2
is proven by Lemma 5.1. Assume that the result is true when the size is k
if m1 , . . . ,2mk
6A. 1
.
and à = 6
4 .
0
1
1. That is,
are each 3integers and Si is a subsemigroup of Mmi (R) for i 2 [1, k 1],
. . . A1k 7
.. 7
..
.
. 5 2 T (S1 , . . . , Sk 1 ) , there exists a weak transfer homomorphism
. . . Ak 1
✓˜ : T (S1 , . . . , Sk 1 ) !
k
Y1
˜ Ã) = (A1 , . . . , Ak 1 ).
Si defined by ✓(
i=1
⇠
By Lemma 4.4, T (S1 , . . . , S2
k) =
"
#
0
00
6A. 1
A A
.
as A =
, where A0 = 6
4 .
0 Ak
0
T (T (S1 , . .3. , Sk 1 ), Sk ). In particular, we can write A
. . . A1k 7
.. 7
..
00
.
. 5 2 T (S1 , . . . , Sk 1 ) and A 2 Mn,mk , where
. . . Ak 1
"
#!
k 1
X
0
00
A A
n=
mi . Define ✓1 : T (T (S1 , . . . , Sk 1 ), Sk ) ! T (S1 , . . . , Sk 1 ) ⇥ Sk by ✓1
=
0 Ak
i=1
(A0 , Ak ). Then, by Lemma 5.1, we know ✓1 !
is a weak transfer homomorphism.
k
k
1
Y
Y
Now, by Lemma 3.4,
Si ⇠
Si ⇥ Sk . Define ✓2 : T (S1 , . . . , Sk 1 ) ⇥ Sk !
=
i=1
i=1
!
k
Y1
˜ 0 ), Ak = (A1 , . . . , Ak 1 ), Ak . By the inductive
Si ⇥ Sk by ✓2 (A0 , Ak ) = ✓(A
i=1
hypothesis, ✓˜ is a weak transfer homomorphism. Furthermore, the identity mapping from
Sk ! Sk is a weak transfer homomorphism and sends Ak to Ak . Thus, by Lemma 3.6, ✓ a
weak transfer homomorphism.
02
31
B6A. 1 .. . . A.1k 7C
6 .
..
.. 7C = (A1 , . . . , Ak ), then
If we define ✓ : T (S1 , . . . , Sk ) !
Si by ✓ B
@4 .
5A
i=1
0 . . . Ak
✓ = ✓2 ✓1 and hence, by Theorem 2.10, ✓ is a weak transfer homomorphism.
k
Y
CHAPTER 5. FACTORING BLOCK TRIANGULAR MATRICES
46
The next corollary shows how the result from Theorem 5.2 is useful.
Corollary 5.3. Let R be a Dedekind-finite ring with unity, k, m1 , . . . , mk 2 N and, for
each i 2 [1, k], let Si be an atomic Dedekind-finite subsemigroup (
of Mmi (R). Let A 2 )
k
X
T (S1 , . . . , Sk ) with diagonal entries A1 , . . . , Ak . Then LT (S1 ,...,Sk ) (A) =
mi : mi 2 LSi (Ai ) .
i=1
Proof. By Theorem 5.2, we have that ✓ : T (S1 , . . . , Sk ) !
k
Y
Si as defined in the theorem
i=1
is a weak transfer homomorphism. Thus, by Theorem 2.6, LT (S1 ,...,Sk ) (A) = L Q
k
Hence, by Remark 3.13, LT (S1 ,...,Sk ) (A) =
(
k
X
i=1
mi : mi 2 LSi (Ai )
)
i=1
Si
(✓(A)).
Chapter 6
Applications and Examples
Since the definition of the generalized block triangular matrix semigroup is so general, the
main result, Theorem 5.2, can be applied to many situations. In this chapter we give some
corollaries of Theorem 5.2 and provide examples of how it can be used in the study of
factorization.
Corollary 6.1. Let n, k1 , k2 , . . . , kn each be integers and let R be a stably finite
02 atomic ring.31
B6A. 1 .. . . A.1n 7C
6 .
..
.. 7C =
The map ✓ : T (Mk1 (R), . . . , Mkn (R)) ! (Mk1 (R), . . . , Mkn (R)) defined by ✓ B
@4 .
5A
0 . . . An
(A1 , . . . , An ) is a weak transfer homomorphism.
Proof. This is simply a restatement of Theorem 5.2 with Si = Mki (R) for each i 2 [1, n].
Corollary 6.1 deals with block triangular matrices with matrices of arbitrary sizes. In
Corollary , we restrict to block triangular matrices of same sized matrices.
47
CHAPTER 6. APPLICATIONS AND EXAMPLES
48
Corollary 6.2. Let n and k be integers
02 and let R 3be
1 a stably finite atomic ring. ✓ :
B6A. 1 .. . . A.1n 7C
n
6 .
..
.. 7C = (A1 , . . . , An ) is a weak transfer
Tn (Mk (R)) ! Mk (R) defined by ✓ B
@4 .
5A
0 . . . An
homomorphism.
Proof. This follows directly from Corollary 6.1 where k1 = k2 = · · · = k.
The following result is proven in [DB14] in the case that R is an integral domain. Even
in a more general setting, this result follows as a corollary to our results of Chapter 5.
Corollary 6.3. Let
n be
02
B6a.1
6.
(R• )n defined by ✓ B
@4 .
0
an integer
31and let R be a stably finite atomic ring. ✓ : Tn (R) !
. . . a1n 7C
. . . .. 7C = (a , . . . , a ) is a weak transfer homomorphism.
. 5A
1
n
. . . an
Proof. This follows directly from Corollary 6.2 where k = 1. Also, it can be viewed as a
direct corollary of Theorem 5.2, where Si = R for each i.
The next corollary provides a useful fact about these types of matrix semigroups.
Corollary 6.4. Let n be an integer and let R be a stably finite atomic ring. Then Tn (R) is
half-factorial if and only if R• is half-factorial.
Proof. This follows directly from Corollaries 6.3 and 3.14.
Now we provide an example that demonstrates Corollaries 6.1 through 6.3 as well as
corollary 5.3.
2
18
6x
Example 6.5. Let P = Z[x , x , x ] . Consider A = 6
4 0
0
3
4
5 •
✓ as in Corollary 6.3, ✓(A) = (x18 , x5 , x13 ).
9
22
3
x x 7
x5 x14 7
5 2 T3 (P). If we define
13
0 x
CHAPTER 6. APPLICATIONS AND EXAMPLES
49
In P, x5 is irreducible, x13 factors only as x13 = (x3 )3 x4 = x3 (x5 )2 = (x4 )2 x5 , and x18
factors only as x18 = (x3 )6 = (x3 )3 x4 x5 = (x3 )(x5 )3 = (x4 )2 (x5 )2 . Thus,
a + b + c : a 2 LP (x18 ), b 2 LP (x5 ), c 2 LP (x13 )
LP 3 (✓(A)) =
= {8, 9, 10, 11}.
Therefore, since ✓ is a weak transfer homomorphism, LT3 (P) (A) = {8, 9, 10, 11}. That is, A
can be writen only as a product of 8, 9, 10, or 11 non-invertible irreducible matrices in T3 (P).
As an example of how the the factorization of elements in T3 (P) relate to those in P 3 ,
we look at the following factorizations of length 8 and 11. Note that
✓(A) = (x18 , x5 , x13 )
= (1, 1, x3 )(1, 1, x5 )2 (1, x5 , 1)(x3 , 1, 1)(x5 , 1, 1)3
= (1, 1, x4 )(1, 1, x3 )3 (1, x5 , 1)(x3 , 1, 1)6 .
Thus, we are able to write A as a product of atoms in T3 (P) whose images are associate to
those atoms in the products above. Using the construction in Lemmas 5.1 and 5.2, we have
the following.
2
18
6x
A = 6
4 0
0
2
61
= 6
40
0
2
61
= 6
40
0
9
0
1
0
0
1
0
22
3
x x 7
x5 x14 7
5
0 x13
32
0 7 61
6
07
5 40
3
x
0
32
0 7 61
6
07
5 40
x4
0
32
0 0 7 61
6
1 07
5 40
5
0 x
0
32 2
0 0 7 61
6
1 07
5 40
0 x3
0
22
32
0 x 7 61
6
1 x14 7
5 40
5
0 x
0
32
0 x22 7 61
6
1 x14 7
5 40
0 x3
0
9
32
3
x 07 6x
6
x5 0 7
54 0
0 1
0
32
x9 0 7 6 x3
6
x5 07
54 0
0 1
0
32
5
0 0 7 6x
6
1 07
54 0
0 1
0
36
0 07
1 07
5 .
0 1
33
0 07
1 07
5
0 1
CHAPTER 6. APPLICATIONS AND EXAMPLES
50
The following result is especially nice, but requires the strong assumption that the underlying ring is a unique factorization domain.
Corollary 6.6. Let n be an0integer
and let
D be a unique factorization domain. Then
2
31
n
B6a.1 .. . . a1n
7C Y
.
•
B
6
7
C
. . ..
✓ : Tn (D) ! D defined by ✓ @4 ..
ai is a weak transfer homomorphism.
5A =
i=1
0 . . . an
• n
Proof.
02 Note that since
31 D is commutative, D is stably finite. Define ✓1 : Tn (D) ! (D ) by
7C
B6a.1 .. . . a1n
6 ..
. . ... 7C = (a1 , . . . , an ). By Corollary 6.3, ✓1 is a weak transfer homomorphism.
✓1 B
@4
5A
0 . . . an
n
Y
• n
•
Define ✓2 : (D ) ! D by ✓2 (a1 , . . . , an ) =
ai . By Theorem 3.9, ✓2 is a weak transfer
i=1
homomorphism. Then by Theorem 2.10, ✓ = ✓2
✓1 is a composition of weak transfer
homomorphisms and is therefore a weak transfer homomorphism.
The following example illustrates the usefulness of Corollary 6.6.
Example 6.7. Consider the polynomial ring Z[x]. This is a unique factorization domain,
h 2
i
x 1 5x2 7x
so we may apply Corollary 6.6. Consider A =
2 T3 (Z[x]). Then if we define
0
1 3
0
✓ as in Corollary 6.6, ✓(A) = 6x
3
6x =
0 6x
2(x + 1)3(1
x)x. Since ✓ is a weak transfer
homomorphism, we are able to write A as a product of 5 atoms in T3 (Z[x]). Using the
construction in Lemma 5.1 and Theorem 5.2, we have the following:
2
2
6x
A = 6
4 0
0
2
61 0
= 6
40 1
0 0
2
1 5x
1
0
32
07 6 1
6
07
5 40
2
0
3
7x7
37
5
6x
32
0 07 61 5x
6
1 07
5 40 1
0 3
0 0
2
32
32
3
7x7 6x + 1 0 07 6x 1 0 07
6
6
37
1 07
1 07
54 0
54 0
5.
x
0
0 1
0
0 0
CHAPTER 6. APPLICATIONS AND EXAMPLES
51
The next two results show that the determinant map is a weak transfer homomorphism
on block triangular matrices if the underlying ring is a principle ideal domain.
Lemma 6.8. Let R be a commutative ring, k, m1 , . . . , mk 2 N, n =
k
X
mk , and for i 2 [1, k],
2
3
k
Y
6A. 1 .. . . A.1k 7
.
..
.. 7 2
Si a subsemigroup of Mmi (R). Then det(A) =
det(Ai ) where A = 6
4 .
5
i=1
0 . . . Ak
i=1
T (S1 , . . . , Sk ).
Proof. We prove by induction on k.2 The case n = 1 3is trivial.
2 Suppose k = 2. Let A = 3
2
3
6a(m1 +1)(m1 +1) . . . a(m1 +1)m2 7
6 a11 . . . a1m1 7
7
A
A
6
7
6
1
12
6
7
..
.. 7
..
..
..
...
6
7,
,
A
=
.
4
5 2 T (S1 , S2 ) where A1 = 6
.
.
.
.
2
6
7
6
7
⇤
4
5
4
5
0
A2
am1 1 . . . am1 m1
am2 (m1 +1)
...
am2 m2
2
3
2
3
6 a1(m1 +1) . . . a1m2 7
6a(m1 +1)1 . . . a(m1 +1)m1 7
6
7
6
7
..
.. 7
..
..
..
..
⇤
7 such that aij = 0 if
6
A12 = 6
,
and
0
=
.
.
.
.
.
.
6
7
6
7
4
5
4
5
am1 (m1 +1) . . . am1 m2
am2 1
...
am2 m1
i > m1 and j  m1 .
mY
1 +m2
X
Then the Leibniz formula states that det(A) =
sgn( )
ai (i) , where Sm1 +m2
2Sm1 +m2
i=1
is the permutation group on [1, m1 + m2 ] and sgn( ) is the sign of the permutation . Note
mY
1 +m2
that if i > m1 and j  m1 , aij = 0. Thus, if i > m1 but (i)  m1 ,
ai (i) = 0.
i=1
Then if
mY
1 +m2
i=1
ai
(i)
6= 0, the permutation
must permute the first m1 indices among them-
selves, and hence must also permute the last m2 indices among themselves. In this case,
Sm1 +m2 ⇠
= Sm1 ⇥ Sm2 , where if
2 Sm1 +m2 corresponds to (⇡, ⇢) 2 Sm1 ⇥ Sm2 ,
=⇡ ⇢
where both ⇡ and ⇢ are both extended to permute all m1 + m2 indicies, ⇡ fixing [1, m1 ] and
⇢ fixing [m1 + 1, m1 + m2 ]. Furthermore, if
corresponds to (⇡, ⇢), sgn( ) = sgn(⇡) sgn(⇢).
CHAPTER 6. APPLICATIONS AND EXAMPLES
52
Thus, since we may ignore the zeros in the sum, we can write
X
det(A) =
sgn( )
ai
(i)
i=1
2Sm1 +m2
=
mY
1 +m2
X
sgn(⇡) sgn(⇢)
(⇡,⇢)2Sm1 ⇥Sm2
=
X
X
sgn(⇡)
=@
X
sgn(⇡)
m1
Y
i=1
⇡2Sm1
ai⇡(i)
i=1
⇡2Sm1 ⇢2Sm2
0
m1
Y
= det(A1 )det(A2 )
10
ai⇡(i) A @
!
m1
Y
ai⇡(i)
i=1
!
sgn(⇢)
m2
Y
m2
Y
ai⇢(i)
i=1
X
sgn(⇢)
m2
Y
i=1
⇢2Sm2
ai⇢(i)
i=1
!
!!
1
ai⇢(i) A
2
3
6A. 1 .. . . A.1n 7
.
..
.. 7 2
Now assume that if A 2 T (S1 , . . . , Sk 1 ), det(A ) =
det(Ai ). Let A = 6
4 .
5
i=1
0 . . . An
T (S1 , . . . , Sk ). Then by Lemma 4.4, T (S1 , . . . ,2Sk ) ⇠
. . , Sk 1 ), Sk ). Thus, we
= T (T (S1 , . 3
"
#
6A. 1 .. . . A1(k. 1) 7
A0 A00
0
.
..
.. 7 2 T (S1 , . . . , Sk 1 ) and
can express A as A =
, where A = 6
4 .
5
0 Ak
0 . . . Ak 1
2
3
6 A.1k 7
0
. 7
A00 = 6
4 . 5. As proven in the case k = 2, det(A) = det(A ) det(Ak ). By the inductive
A(k 1)k
0
hypothesis, det(A0 ) =
k
Y1
0
k
Y1
det(Ai ). Thus,
i=1
det(A) =
k
Y1
i=1
!
det(Ai ) det(Ak ) =
k
Y
i=1
det(Ai ).
CHAPTER 6. APPLICATIONS AND EXAMPLES
53
We are now able to prove the following.
Theorem 6.9. Let n, k1 , . . . , kn be integers and let D be a principle ideal domain. Then
det : T (Mk1 (D), . . . , Mkn (D)) ! D• is a weak transfer homomorphism.
31
02
B6A. 1 .. . . A.1n 7C
6 .
..
.. 7C =
Proof. Define ✓1 : T (Mk1 (D), . . . , Mkn (D)) ! (Mk1 (D), . . . , Mkn (D)) by ✓1 B
@4 .
5A
0 . . . An
(A1 , . . . , An ). By Corollary 6.1, ✓1 is a weak transfer homomorphism.
Define ✓2 : (Mk1 (D), . . . , Mkn (D)) ! (D• )n by ✓2 (A1 , . . . , An ) = det(A1 ), . . . , det(An ) .
As shown in [DA11], for each i 2 [1, n], det : Mki (D) ! D• is a transfer homomorphism.
Thus, by Theorem 2.8, the determinant is a weak transfer homomorphism. Hence, by Theorem 3.7, ✓2 is a weak transfer homomorphism.
• n
•
Define ✓3 : (D ) ! D by ✓3 (a1 , . . . , an ) =
n
Y
ai . Since D is a principle ideal do-
i=1
main, D is a unique factorization domain. Thus, by Theorem 3.9, ✓3 is a weak transfer
homomorphism.
Lastly, by Theorem 6.8, det(A) =
n
Y
det(Ai ). That is, det(A) = (✓3 ✓2 ✓1 )(A). Thus,
i=1
the determinant map is a composition of weak transfer homomorphisms and, by Theorem
2.10, is a weak transfer homomorphism from T (Mk1 (D), . . . , Mkn (D)) to D• .
The diagram below shows a representation of the composition of maps used in the proof
above to show the determinant map is a weak transfer homomorphism.
T (Mk1 (D), . . . , Mkn (D))
✓1
/
Mk1 (D) ⇥ · · · ⇥ Mkn (D)
✓2
/
(D• )n
✓3
det
The following example shows how the weak transfer homomorphism from Theorem 6.9
can be used.
/ 2 D•
CHAPTER 6. APPLICATIONS AND EXAMPLES
2h
6
Example 6.10. Let A = 4
2 5
6 9
0
i ⇥
17
19
9 11
⇥
8 11
18 23
⇤
54
3
7
3
⇤5 2 T3 M2 (Z) . Then det(A) = 168 = 2 · 3 · 7.
Thus, since the determinant map is a weak transfer homomorphism, L
T3 M2 (Z)
(A) = {5}.
That is, A can only be written as a product of 5 irreducible matrices. We show two such
factorizations.
A =
=
i  I h 17 19 i
2
h 0 i
9 11
2 1
0
5]
0 [3
4 3
" 
#  h2 1 i
18 2
2 1
I2
I2
1 3
h12 27i
h
i
0 7
5 8
0
0
1 3
11 18
h I2
h
[ 24 35 ]
0
h
0
I2
ihh
7 12 ]
[ 13
22
0
0
I2
3 1
11 3
0
i
i
0
I2
i
69 95
43 61
0
5 2 0
19 7
0
I2

47 17
0
33 12
I2
0
0
I2
The next example shows that the result in Theorem 6.9 is not generally true if the
underlying ring is not a principle ideal domain.
Example 6.11. Consider the following matrix in S = T2 (M2 (Z[x, y])).
2h
6
A=4
It is shown in [EM79] that the matrix
x2 xy 2
xy+2 y 2
[ 00 00 ]
h
3
i
x2 xy 2
xy+2 y 2
i
[ xx xx ]7
5
[ 10 01 ]
is an atom in M2 (Z[x, y]). Thus, by
Theorem 4.8, we know A is an atom of S and so LS (A) = {1}. However, det(A) = 4 = 22 .
Thus, LZ[x,y] (det(A)) = {2}, and so LS (A) 6= LZ[x,y] (det(A)). Hence, the determinant map
cannot be a weak transfer homomorphism.
In Corollary 6.6, we saw that if D is a unique factorization domain, Tn (D) has the same
factorization properties as D• . As shown in Example 6.11, this is not true for a general ring.
However, we do have the following result as consolation. This result parallels Theorem 3.11,
and the proof is nearly identical.
CHAPTER 6. APPLICATIONS AND EXAMPLES
Theorem 6.12.
02Let
B6a.1
6.
is defined by ✓ B
@4 .
0
2
55
R be an 3
atomic
Dedekind-finite ring and let n 2 N. If ✓ : Tn (R) ! R•
1
n
. . . a1n 7C Y
.. 7C
..
ai , then LTn (R) (A) ✓ LR• ✓(A) .
. . 5A =
i=1
. . . an
3
6a.1 .. . . a1n
7
..
. . ... 7 2 Tn (R) and m 2 LTn (R) (A). Then there exist atoms
Proof. Let A = 6
4
5
0 . . . an
2
3
6b.i1 .. . . bi1n
7
..
. . ... 7 and A = B1 · · · Bm . Since Bi is an
B1 , . . . , Bm of Tn (R), where Bi = 6
4
5
0 . . . ain
atom for each i 2 [1, m], by Lemma 4.8, there exists hi 2 [1, m] such that bihi is an atom of
m
Y
R and bij 2 R⇥ for each i 2 [1, m]\{hi }. Furthermore, for each i 2 [1, m], ai =
bji .
j=1
Then ✓(A) = a1 · · · an = (b11 · · · bm1 ) · · · (b1n · · · bmn ). This product has m atoms, namely
bihi for each i 2 [1, m], with the remaining elements units. That is, bij 2 (R• )⇥ if (i, j) 2
/
{(i, hi ) : i 2 [1, m]}. Thus, by Lemma 1.12, ✓(A) can be written as a product of m atoms,
and m 2 LR• ✓(A) .
Note that Theorem 6.12 could also be proven as a corollary of Theorems 3.11 and 6.3.
As we did with Theorem 3.11, we provide an example demonstrating Theorem 6.12 and the
failure of its converse.
Example 6.13. Let P = Z[x3 , x4 , x5 ]• and let A =
⇥ x4
x5
0 x6
⇤
2 T2 (P). The only way to factor
⇥
⇤⇥ 5⇤⇥
⇤
A as a product of atoms, up associates and a permutation, is A = 10 x03 10 xx3 x04 01 . Thus,
LT2 (P) (A) = {3}. If we denote ✓ as in Theorem 6.12, we have ✓(A) = x10 = (x3 )2 x4 = (x5 )2 .
Thus, LP (✓(a)) = {2, 3}. As can be seen, LT2 (P) (A) ⇢ LP (✓(A)), but LP (✓(A)) 6⇢ LT2 (P) (A).
In Theorem 6.9, we constructed a weak transfer homomorphism from a block triangular
matrix semigroup to its underlying ring. The weak transfer homomorphism greatly simplifies
the study of factorizations in semigroups of generalized block triangular matrices when the
CHAPTER 6. APPLICATIONS AND EXAMPLES
56
underlying ring is a principal ideal domain. When the underlying ring is not a principal ideal
domain, we can still sometimes map the block triangular matrix to much nicer rings. The
next corollary and example show how Theorems 5.2 and 3.7 can be used in combination to
send elements from one semigroup to a much nicer one.
Corollary 6.14. Let R be a stably finite atomic ring, k, m1 , . . . , mk 2 N, and for each i 2
[1, k], let Si be a subsemigroup of Mmi (R). If there exists a weak transfer
31 i :
02homomorphism
Si ! Ti for each i 2 [1, k], then ✓ : T (S1 , . . . , Sk ) !
1 (A1 ), . . . ,
k (Ak )
B6A. 1 .. . . A.1k 7C
6 .
..
.. 7C =
Ti defined by ✓ B
@4 .
5A
i=1
0 . . . Ak
k
Y
is a weak transfer homomorphism.
02
31
k
Y
B6A. 1 .. . . A.1k 7C
6 .
..
.. 7C = (A1 , . . . , Ak ). By TheoProof. Define ✓1 : T (S1 , . . . , Sk ) !
Si by ✓ B
@4 .
5A
i=1
0 . . . Ak
rem 5.2, ✓1 is a weak transfer homomorphism. Define ✓2 :
k
Y
i=1
1 (A1 ), . . . ,
k (Ak )
Si !
k
Y
Ti by ✓2 (A1 , . . . , Ak ) =
i=1
. By Theorem 3.7, ✓2 is a weak transfer homomorphism. Thus, by The-
orem 2.10, since ✓ = ✓2
✓1 , ✓ is a composition of weak transfer homomorphisms and is
therefore a weak transfer homomorphism.
⇣
⌘
Example 6.15. Let P = Z[x3 , x4 , x5 ]• and T = T U T2 (P), M2 (Z), LT3 (P) , where U T2 (P)
is the semigroup of 2 ⇥ 2 upper triangular matrices, Mn (Z) is the semigroup of 2 ⇥ 2 integer
matrices, and LT3 (P) is the semigroup of 3 ⇥ 3 lower triangular matrices. By Corollary 6.3,
the map ✓1 : U T2 (P) ! P 2 defined by ✓1 ([ a0 cb ]) = (a, b) is a weak transfer homomorphism.
By Theorem 6.9, ✓2 = det : M2 (Z) ! Z• is a weak transfer homomorphism. By an argument
⇣h a 0 0 i⌘
similar to that found in Corollary 6.3, the map ✓3 : LT3 (P) ! P defined by ✓ d b 0
=
f e c
(a, b, c) is a weak transfer homomorphism. Hence, by Corollary 6.14, ✓ : T ! P 2 ⇥ Z• ⇥ P 3
as defined in the theorem is a weak transfer homomorphism.
CHAPTER 6. APPLICATIONS AND EXAMPLES
2h
6
Consider A = 6
4
x15 x9
0 x4
0
i 3
i h 3 4
x2 x3
x x 1
9
7
4
6
10
x x
h x5 x12x 3 i
7
x x x
[ 87 10
]
7
9 1 x3
8
x
" 10
#5
x
0 0
0
x5 1 0
x4 x9 x15
i h
0
57
2 T . By Theorem 2.6, since ✓ is a weak
transfer homomorphism, LT (A) = LP 2 ⇥Z• ⇥P 3 (✓(A)). Therefore, we may study the factorizations of A by considering ✓(A). We know ✓(A) = (x15 , x4 , 6, x10 , 1, x15 ). Hence, by Remark
3.13, the factorization lengths of ✓(A) depends on the factorization lengths of each of its
entries in their individual semigroups. We have the following:
5
2
3
3
• LP (x15 ) = {3, 4, 5} since x15 = (x3 ) = (x3 ) x4 x5 = (x3 )(x4 ) = (x5 ) are the only
factorizations of x15 as a product of atoms, up to associates and permutations.
• LP (x4 ) = {1} since x4 is an atom.
• LZ• ( 6) = {2} since
6 = 2( 3) and Z is half-factorial.
• LP (x10 ) = {2, 3} since x10 = (x3 )2 x4 = (x5 )2 are the only factorizations of x10 as a
product of atoms, up to associates and permutations.
• LP (1) = {0} since 0 is a unit.
Therefore, by Remark 3.13,
LT (A) =
(
6
X
i=1
mi : m1 , m6 2 LP (x15 ), m2 2 LP (x4 ), m3 2 LZ ( 6), m4 2 LP (x10 ), m5 2 LP (1)
= {3, 4, 5} + {1} + {2} + {2, 3} + {0} + {3, 4, 5}
= {11, 12, 13, 14, 15, 16}
)
CHAPTER 6. APPLICATIONS AND EXAMPLES
58
We show two factorization of A, one of maximal length and the other of minimal length.
A =
" I2
0
0
0 I2  0
x5 0 0
0 0
0 1 0
0 0 1
"
I
0
0
2] 0
0 [0
1 3
0
0 I3
2
=
" I2
0
0
0 I2  0
x3 0 0
0 0
0 1 0
0 0 1
"I
 0
2 4
0
1 3
0
0
2
0
0
I3
# 2 I2
4
0
0 I2 "
0 0
i
x2 x3 0
9
7
hx x i
5 7
0
0
4 5
0
0
I3
I2
h
3
# 2 2 I2
#"
4
I2
0
0
h
0
0 I2 "
0 0
x2 x3
x9 x7
[ 23 13 ]
0
i3
x3 x4 1
4
6
10
x x x
6 0 I2 h x59 x12 x33 i 7
4
x
 1 x 5
1 0 0
0 1 0
0 0
0 0 x5
 h x5 0 i
3
0 0
0 1
0
I2 0
0
0 I3
" I2 0
0
0
0
0 I2  0
#
x5 0 0 5
1 0 0
0 0 0 1 0
x5 1 0
0 0 x5
x4 x9 1
i
#h
1 x9
0 x4
0
0
i
0 0
I2 0
0 I3
3
0
I3
#h
1 x9
0 x4
0
0
i
0 0
I2 0
0 I3
2
I2 0
2
h
i3
x3 x4 1
4 x6 x10
x
6 0 I2 h x59 x12 x33 i 7
4
x
 1 x 5
1 0 0
0 1 0
0 0
0 0 x3
 h x3 0 i
5
0 0
0 1
0
I2 0
0
0 I3
" I2 0
0
0
0
0 I2  0
#5
4
x 0 0
1 0 0
0 0 0 1 0
x5 1 0
0 0 x3
x4 x9 1
0
#2
#4
I2 0
h
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