IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEXES. Jake Hobson

Transcription

1 IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEXES by Jake Hobson 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, 2012

2 ABSTRACT by Jake Hobson Recently there has been much research on irreducible divisor graphs visual representations of the factorizations of elements in commutative rings. In this thesis we expand on this concept by introducing the irreducible divisor simplicial complex of an element in an integral domain, effectively constructing a higher dimensional analog of the irreducible divisor graph. We show that this new construction often sheds more light on the factorization of elements than its two-dimensional counterpart. In addition, we generalize several important irreducible divisor graph results.

3 IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEXES by Jake Hobson 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, 2012

4 IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEXES by Jake Hobson APPROVED: Committee Member esis Committee Member ACCEPTED: Chair, Department of Mathematics and Computer Science Dean. Graduate School UNIVERSITY OF CENTRAL MISSOURI WARRBNSBURG, MISSOURI

5 ACKNOWLEDGMENTS I would first like to thank my fiancé Courtney for allowing me to work on this thesis for a finite, but seemingly uncountable number of evenings that I should have been spending with her! Also, I would like to thank my thesis advisor and mentor Dr. Nicholas Baeth for having the patience to teach me how to be a mathematician. Last, I would like to thank those in the Math department at UCM who I have been lucky enough to work with. Without your flexibility and understanding I certainly would not have been able to pursue my math dreams at all.

6 Contents 1 Introduction Motivation Preliminaries Algebraic Definitions Examples Graphs Simplicial Complexes Factorization studies Irreducible Divisor Graphs The Irreducible Divisor Simplicial Complex Introduction Main Results Elasticity Conclusion Bibliography 34 vi

7 List of Figures 1.1 A connected but non-complete graph A complete graph with a loop The simplicial complex S = (V, F ) K 1 (S), the 1-skeleton of S K 0 (S), the 0-skeleton of S The simplicial complex T The join of two simplicial complexes: S T The simplicial complex T U G(18) in Z[ 5] G(108) in D = Z[ 5] S(18) in Z[ 5] S(108) in Z[ 5] G(x 17 ) in F[x 4, x 5, x 6, x 7 ] S(x 17 ) in F[x 4, x 5, x 6, x 7 ] vii

8 Chapter 1 Introduction 1.1 Motivation Studies in factorization theory have been prevalent in much of modern mathematics, with deep roots going back as far as Euclid s monumental Fundamental Theorem of Arithmetic. This theorem tells us that every positive integer factors uniquely (up to order) as a product of primes. Moreover, this uniqueness of factorization extends to all of Z: every integer factors uniquely up to order and up to associates as a product of primes. For example 2100 = = ( 2) 2 ( 3)( 5) 2 ( 7), but we do not consider these to be distinct factorizations since we have only replaced each prime factor p with its additive inverse p. However, in many algebraic settings factorization is not unique even with this restrictive definition of unique. Perhaps the most frequently mentioned ring with non-unique factorization is Z[ 5] = {a + b 5 : a, b Z} where 6 factors as both 2 3 and as (1 + 5)(1 5). We will return to this ring several times throughout this thesis. Much research has been done on factorization theory over the past 50 years. Perhaps one of the most comprehensive texts is provided by Halter-Koch and Geroldinger [6]. 1

9 CHAPTER 1. INTRODUCTION 2 In [5] Coykendall and Maney introduce the concept of an irreducible divisor graph. In this paper they investigate a way to represent the factorizations of an element x in an integral domain as a graph, where the vertices are a pre-chosen set of irreducible divisors of x and an edge connects two vertices if and only if the corresponding irreducible divisors appear in the same factorization of x. In both [2] and [5] the utility of the irreducible divisor graph is illustrated when it is shown that an integral domain D is a unique factorization domain if and only if each irreducible divisor graph is complete, i.e. all possible edges exist. Since then, irreducible divisor graphs have been studied in the context of integral domains in [3], [7], and [5], and in more general contexts in [1], [4], [8], and [9]. Despite the appealing result mentioned above, difficulties arise when considering irreducible divisor graphs of elements having two or more factorizations each involving one specific irreducible element (e.g. x factors as x = ab and as x = cb where b is irreducible and a c). In short, irreducible divisor graphs fail to give us all the information we might wish to glean about an element s factorizations. We will discuss these difficulties and provide a way to mitigate some of them. The main goal of this paper is to introduce the concept of an irreducible divisor simplicial complex. This is effectively a generalization of the irreducible divisor graph to higher dimensions. As we shall see, irreducible divisor simplicial complexes often convey more information about the factorization of an element than its two-dimensional counterpart. In [7] Maney uses homologies to study irreducible divisor graphs. In his exploration he links irreducible divisor graphs to the zeroth and first homologies (which are defined in [7] and shall not be discussed here). Maney goes further to study higher homologies which, although they are not explicitly mentioned in [7], are related to irreducible divisor simplicial complexes. This gives yet another motivation for studying this new construct that we will define in Section 3.1.

10 CHAPTER 1. INTRODUCTION 3 In this chapter we will provide some preliminary definitions chiefly related to factorization. We also give a primer in graph theory and define simplicial complexes and related terminology. In Chapter 2 we will share already developed results regarding irreducible divisor graphs. In Chapter 3 we introduce irreducible divisor simplicial complexes and extend previous results to our new construction. 1.2 Preliminaries In this section we introduce basic terminology and definitions that will be used throughout this thesis Algebraic Definitions We begin with a brief review of common algebraic definitions. Definition A ring is a set R together with two binary operations, typically addition (+) and multiplication ( ), such that (R, +) is an abelian group and multiplication is distributive over addition. Throughout, we assume that all rings are commutative with respect to multiplication and have unity (multiplicative identity) 1 0. We say that an element a of a commutative ring R is a zero divisor if there exists a non-zero element b in R such that ab = 0. An element a in R is said to be a unit in R if there is some b in R with ab = ba = 1. We denote the set of non-zero elements of a ring R as R and the set of units of R as U(R). It will often be convenient for us to speak of the set of non-zero non-units of R, which is denoted as R \U(R). Definition A commutative ring D is an integral domain if it contains no non-zero zero divisors.

11 CHAPTER 1. INTRODUCTION 4 It is important to remember that multiplication in integral domains is a cancellative operation. That is, if a, b, c D with a 0 and ab = ac, then b = c. We now justify this statement. Let a, b, c D. If ab = ac, then a(b c) = 0. Since D contains no non-zero zero divisors and a 0, b c = 0, and thus b = c. The following definitions will be important as we study the factorization of elements in integral domains. Definition Let D be an integral domain. 1. We say x D \U(D) is irreducible (or an atom) if whenever x = yz with y, z D, then either y U(D) or z U(D). 2. We say x D \U(D) is prime if whenever x yz with y, z D, then either x y or x z. 3. We say an element x D is square free if it is not divisible by any perfect square z 2 D \U(D), that is, if z 2 x for z D, then z U(D). 4. Elements a and b of D are called associates if a = ub where u U(D). Note that the relation a b on elements of D is an equivalence relation that partitions D into associate classes. 5. We denote the set of irreducibles in D as Irr(D) = {x : x is irreducible}. We define Irr(D) to be a (pre-chosen) set of associate class representatives, one from each class of non-zero associates. We denote the irreducible divisors of a particular element x D as Irr(x) and set Irr(x) = Irr(x) Irr(D). 6. We say that D is a unique factorization domain (UFD) if every non-zero non-unit x in D factors uniquely as a product of irreducibles.

12 CHAPTER 1. INTRODUCTION 5 We now introduce a foundational concept; that of an atomic domain. Definition We say that an integral domain D is atomic if each element x D \U(D) can be factored into a finite product of irreducible elements. If we attempt to investigate factorization in non-atomic domains we encounter significant problems. The following example illustrates this struggle. Example Consider the ring A = Q a 12 x a32 x apq x p q + : aij [ {x m } ] n = {a m,n N 0 + a 11 x + a 21 x 2 + a 31 x Q, and only finitely many coefficients are non-zero}. We note that the only units of A are the elements of Q. Consider the non-zero non-unit x A. Factoring x, we have x = x 1 2 x 1 2 = x 1 2 x 1 4 x 1 4 = x 1 2 x 1 4 x 1 8 x 1 8 = In fact, for each positive integer n, x 1 n is a non-zero non-unit that is not irreducible since x 1 n = x 1 2n x 1 2n =. Thus x cannot be written as a finite product of irreducibles, so A is not atomic. It is easy to see that an attempt to investigate factorizations in this domain will be met with much frustration because one can never easily write down the factorization of a particular element. It is known that in any integral domain all primes are irreducible. To wit, if x is prime in an integral domain D with x = yz then x y or x z. Without loss of generality suppose xt = y for some t D. Then x = yz = (xt)z = x(zt). Canceling yields 1 = tz which implies that z is a unit. Therefore x is irreducible. If D is an atomic domain the converse is equivalent to D being a UFD. Proposition [6, Theorem ] An atomic domain D is a UFD if and only if all irreducibles in D are prime.

13 CHAPTER 1. INTRODUCTION 6 In light of Example 1.2.5, we will restrict our attention to atomic domains throughout the remainder of this thesis. Definition Suppose D is an atomic domain. 1. We say D is a finite factorization domain (FFD) if every non-zero non-unit in D has only finitely many distinct non-associate irreducible divisors. 2. Let D be a finite factorization domain. The set of lengths (of factorizations) of x D \U(D) is L(x) = {t : x = a 1 a 2 a t where each a i is irreducible}. 3. We say D is a bounded factorization domain (BFD) if there is a bound on the length of factorization into products of irreducible elements for each non-zero non-unit element in D. In other words, L(x) is a bounded function. 4. If s = t whenever x = a 1 a s = b 1 b t for irreducibles a i, b j D, we say that D is a half-factorial domain (HFD). Note that D is a HFD if an only if L(x) is a singleton for each x D \U(D). We can also see the following implications: UFD FFD BFD and UFD HFD BFD. There are several invariants that are commonly studied in factorization theory. We now define one particularly interesting invariant that we will investigate by using simplicial complexes. Definition Let D be a bounded factorization domain. We define the elasticity of an element x D \U(D) as ρ(x) = max(l(x))/min(l(x)). The elasticity of D is then defined to be ρ(d) =sup{ρ(x) : x D \U(D)}.

14 CHAPTER 1. INTRODUCTION 7 The elasticity of an element x gives us a measure of how far x is from having unique factorization. All elements in a UFD or HFD have elasticity 1. Indeed, if D is a HFD, and x D \U(D) then L(x) is a singleton, say {a}. Thus ρ(x) = max(l(x))/min(l(x)) = Examples To illustrate the many definitions provided earlier in this chapter we now examine the integral domain Z[ 5] = {a + b 5 : a, b Z}. We will use Z[ 5] repeatedly throughout the text to illustrate various properties of irreducible divisor graphs and irreducible divisor simplicial complexes. When studying factorizations in Z[ 5] a particularly useful tool is the norm, which is the function N : Z[ 5] N defined by N(a + b 5) = a 2 + 5b 2 with the following properties: 1. N(x) = 0 if and only if x = N(xy) = N(x)N(y) for all x, y Z[ 5]. 3. An element x is a unit if and only if N(x) = If N(x) is prime, then x is irreducible in Z[ 5]. Note, the converse of 4 does not always hold. We now verify these properties. Property 1 is immediately obvious since a 2, b 2 0 for all a, b Z. For property 2, let x = a + b 5 and y = c + d 5 be elements of Z[ 5].

15 CHAPTER 1. INTRODUCTION 8 Then N(xy) = N((a + b 5)(c + d 5)) = N(ac + ad 5 + bc 5 5bd) = N((ac 5bd) + (ad + bc) 5) = (ac 5bd) 2 + 5(ad + bc) 2 = a 2 c 2 10abcd + 25b 2 d 2 + 5a 2 d abcd + 5b 2 c 2 = a 2 c 2 + 5a 2 d 2 + 5b 2 c b 2 d 2 = (a 2 + 5b 2 )(c 2 + 5d 2 ) = N(x)N(y). For property 3, first note that if N(a+b 5) = 1, then a 2 +5b 2 = 1. This can only occur if a + b 5 = 1 or a + b 5 = 1, which are clearly units of Z[ 5]. Now, if x is a unit of Z[ 5], then there exists y Z[ 5] such that xy = 1. Thus 1 = N(xy) = N(x)N(y), and thus N(x) = 1. Therefore x = ±1. Property 4 follows immediately from Properties 2 and 3: Suppose N(x) is prime and write x = yz. Then N(x) = N(y)N(z) and since N(x) is prime and Z is a UFD, either N(y) or N(z) is 1 and hence either y or z is a unit of Z[ 5].

16 CHAPTER 1. INTRODUCTION 9 Example Consider the element 6 in Z[ 5]. We have N(6) = Thus if x 6 in Z[ 5], then N(x) {2, 3, 4, 6, 12, 18, 36}. If x = a + b 5, then N(x) = a 2 + 5b 2 and hence we can eliminate most of the above possibilities for N(x): a 2 + 5b 2 = 2 (impossible). a 2 + 5b 2 = 3 (impossible). a 2 + 5b 2 = 4 a = ±2b = 0. a 2 + 5b 2 = 6 a = ±1, b = ±1. a 2 + 5b 2 = 9 a = ±3, b = 0 or a = ±2, b = ±1. a 2 + 5b 2 = 12. If b = 0 then a 2 = 12 or, if b = 1 then a 2 = 7, (both impossible). a 2 + 5b 2 = 18. If b = 0 then a 2 = 18 or, if b = 1 then a 2 = 13, (both impossible). We needn t consider a 2 + 5b 2 = 36 since if N(x) = 36, then x = 6 and 6 is not irreducible. If N(x)=4, x = ±2. If N(x) = 6, x = ±1 ± 5. If N(x) = 9, x = ±3 or x = ±2 ± 5 (note that this last irreducible does not divide 6). This shows us that ±2, ±3, ±1 ± 5 are the only irreducible divisors of 6. Thus the only factorizations of 6 in Z[ 5] are: 6 = 2 3 = (1 + 5)(1 5). Therefore L(6) = {2} and ρ(6) = 2 2 = 1. Example Consider the element 18 in Z[ 5]. Here we have the norm of 18 is N(18) = Thus if x 18 in Z[ 5] then, N(x) {2, 3, 4, 6, 9, 27, 36, 54, 81, 108, 162, 324}. If x = a + b 5, then N(x) = a 2 + 5b 2 and hence we can eliminate most of the above

17 CHAPTER 1. INTRODUCTION 10 possibilities for N(x). For example a 2 + 5b 2 = 12 is impossible. Indeed, if a 2 + 5b 2 = 12, then 4 < a < 4 and 2 < b < 2. Checking the 7 3 = 21 possibilities, we see there are no integers a, b Z with a 2 + 5b 2 = 12. Similarly we eliminate most other cases to show that N(x) {4, 6, 9}. If N(x) = 4, x = ±2. If N(x) = 6, x = ±1 ± 5. If N(x) = 9, x = ±3 or x = ±2 ± 5. We now see that ±2, ±3, ±1 ± 5, ±2 ± 5 are the only irreducible divisors of 18. Thus the only factorizations of 18 in Z[ 5] are: 18 = = 3(1 + 5)(1 5) = 2(2 + 5)(2 5). We see that L(18) = {3} so, ρ(18) = 3 3 = 1. To further our understanding we study the factorization of 108 Z[ 5]. Example First we see that 108 factors into 108 = Using a norm argument we find that the factors of 108 are: 108 = = αα = 2 2 3ββ = 3α 2 α 2 = 2ααββ, where α = 1 + 5, β = and α, β denote their complex conjugates. Therefore L(108) = {5} and ρ(108) = max(l(108))/min(l(108)) = 5 5 = 1. In fact, it is known that Z[ 5] is an HFD i.e., ρ(z[ 5]) = 1.

18 CHAPTER 1. INTRODUCTION Graphs In this section we present some key graph theoretic definitions to which we will refer throughout. Definition A graph is an ordered pair of sets (V, E), where V is called the vertex set, and E is the edge set whose elements are subsets of V of cardinality 2. We denote an edge between vertices a and b as {a, b} and note that the edge {a, b} is the same as the edge {b, a}. The edge {a, b} is said to be incident with both vertices a and b. We denote the set of vertices of a graph G as V (G) and the set of edges of G as E(G). In addition, we define a loop to be an edge between a and itself. Definition Let G be a graph. 1. A walk on G is an alternating series of vertices and edges, beginning and ending with a vertex, in which each edge is incident with the vertex immediately preceding it and the vertex immediately following it. 2. The graph G is connected if there exists a walk between each pair of distinct vertices. 3. We say that G is complete if for all x, y V (G) with x y we have {x, y} E(G). Example The graph G with vertex set V (G) = {a, b, c, d} and edge set E(G) = {{a, b}, {b, d}, {c, b}}, is a connected but not complete graph as pictured in Figure 1.1. The graph H given by vertex set V (H) = {a, b, c, d} and edge set E(H) = {{a, b}, {a, c}, {a, d}, {b, b}, {b, c}, {b, d}, {c, d}} is complete with a loop on the vertex b and is illustrated in Figure 1.2.

19 CHAPTER 1. INTRODUCTION 12 a b c Figure 1.1: A connected but non-complete graph d a b c Figure 1.2: A complete graph with a loop d Simplicial Complexes In this section we define simplicial complexes which can be thought of as higher dimensional analogs of graphs. Definition A simplicial complex S is a pair (V, F ) where V is a set of vertices and F is a collection of subsets of V satisfying: (1) {v} F for all v V (vertices are simplices) and (2) If σ F and τ σ, then τ F (τ is called a face of σ). We denote the set of vertices of S as V (S), and the set of faces of S as F (S). The dimension of a face β of finite cardinality in a simplicial complex S is one less than its cardinality and is denoted as dim(β) = β 1. Just as in graphs, zero dimensional faces are represented by points, and one dimensional faces are represented as edges. Graphically, we illustrate 2-dimensional faces by shaded

20 CHAPTER 1. INTRODUCTION 13 triangles as shown in Figure 1.3. We also represent 3-dimensional faces by solid tetrahedra. For higher dimensional faces the reader is on their own. Definition The maximal faces (with respect to set containment) of a simplicial complex S are called the facets of S. In our studies it will often be useful to consider sub-simplices of simplicial complexes. A sub-simplex T of a simplicial complex S = (V, F ) is a pair (W, H) with W V, H F with H P(W ) such that H meets conditions (1) and (2) from Definition One particularly useful sub-simplex is the k-skeleton. Definition Let k be a non-negative integer. The k-skeleton of a simplicial complex S is the sub-simplex of S consisting of all the faces of S whose dimension is at most k. We denote the k-skeleton of a simplicial complex S as K k (S). Note that if S is a simplicial complex, K 1 (S) is a graph as defined in Definition Example Let S = (V, F ) be the simplicial complex with vertex set V (S) = {a, b, c, d} and face set F (S) = {, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {b, c}, {a, b, c}}. The 0- dimensional faces are the singleton sets {a}, {b}, {c}, and {d}. These singletons are represented graphically as vertices. The 1-dimensional faces are the tuples {a, b}, {a, c}, and {b, c}. The 2-dimensional face is {a, b, c}. The faces {a, b, c} and {d} are the only facets of S. We see that the simplicial complex S 1 formed by V (S 1 ) = {a, b, c} and F (S 1 ) = {, {a}, {b}, {a, b}, {a, c}, {b, c}, {a, b, c}} forms a sub-simplex of S. Note that F (S 1 ) = P({a, b, c}), the power set of V (S 1 ). We will return to this specific kind of complex later. If we consider the 1-skeleton of the simplicial complex S from Example , we are left with only the faces of dimension 0 and 1: F (K 1 (S)) = {, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {b, c}}, represented graphically in Figure 1.4. If we consider the 0-skeleton of S we have F (K 0 (S)) = {, {a}, {b}, {c}, {d}} which gives us the sub-simplex of S shown in Figure 1.5.

21 CHAPTER 1. INTRODUCTION 14 a b c d Figure 1.3: The simplicial complex S = (V, F ) a b c d Figure 1.4: K 1 (S), the 1-skeleton of S a b c d Figure 1.5: K 0 (S), the 0-skeleton of S A useful binary operation on the set of simplicial complexes is called the join. This operation allows us to build new simplicial complexes from old. Definition The join of two simplicial complexes X and Y, denoted X Y, has vertex set V (X Y ) = V (X) V (Y ) and face set F (X Y ) = {A B : A F (X) and B F (Y )}.

22 CHAPTER 1. INTRODUCTION 15 Example Let S be the simplicial complex in Example and let T be the simplicial complex defined by V (T ) = {{a}, {b}, {c}, {d}} and F (T ) = {, {a}, {b}, {c}, {d}, {b, d}} as represented in Figure 1.6. Then S T is defined as V (S T ) = {{a}, {b}, {c}, {d}} and F (S T ) = {, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}. Graphically, S T is shown in Figure 1.7, where the three 2-dimensional faces are shaded and the interior of the tetrahedron constitutes the 3-dimensional facet {a, b, c, d}. a b c d Figure 1.6: The simplicial complex T a d c b Figure 1.7: The join of two simplicial complexes: S T

23 CHAPTER 1. INTRODUCTION 16 Example Let U be the simplicial complex defined by V (U) = {e, f} and F (U) = {, {e}, {f}}, and let T be the simplicial complex defined in Example Then T U is defined by V (T U) = {a, b, c, d, e, f} and F (T U) = {, {a}, {b}, {c}, {d}, {e}, {f}, {a, e}, {a, f}, {b, e}, {b, f}, {c, e}, {c, f}, {d, e}, {d, f}, {b, d, e}, {b, d, f}}. This complex is represented graphically in Figure 1.8. Note the shaded regions representing the two dimensional faces {b, d, f} and {b, d, e}. a b f e c d Figure 1.8: The simplicial complex T U It is worth noting that in the definition of the join X Y of two simplicial complexes some authors require the intersection of the vertex sets V (X) and V (Y ) to be empty. We do not observe this convention as it will often be convenient for us to consider cases where V (X) V (Y ). We now present a lemma regarding the join of simplicial complexes of a specialized type which will be useful later. First recall that if X is a set, then P(X) denotes the power set of X consisting of all subsets of X. We abuse notation and write P(X) to denote the simplicial complex (X, P(X)) with vertex set X and face set P(X). Recall that V (P(X)) = X and F (P(X)) = P(X).

24 CHAPTER 1. INTRODUCTION 17 Lemma Let A and B be two sets. As simplicial complexes, P(A B) = P(A) P(B). Proof. First we show that the vertex sets are equal. Suppose a V (P(A B)). Then a A B, which by definition means a V (P(A) P(B)). For the other containment suppose b V (P(A) P(B)). By definition, b A B and hence b V (P(A B)). Now we show P(A B) and P(A) P(B) have the same face set. Let α F (P(A B)), i.e. α A B. Set α A := α A A and α B := α\α A B. Clearly α = α A α B and hence α F (P(A) P(B)). To show the other containment, select α F (P(A) P(B)) and write α = α A α B for some α A A, α B B. Then α A B and thus α F (P(A B)). Since P(A B) and P(A) P(B) have the same vertex and face sets, they are equal as simplicial complexes.

25 Chapter 2 Factorization studies 2.1 Irreducible Divisor Graphs In this section we introduce the irreducible divisor graph of an element in an atomic domain. This graphical structure gives us a way to visualize the factorization of the element as a product of irreducibles and has been studied in [5] and [2]. It has also been generalized to other contexts in [1], [4], [8], and [9]. We first give the formal definition and then provide a few examples. These examples will illustrate the usefulness of irreducible divisor graphs as well as highlight some of their shortcomings. Definition Let D be an atomic domain and let x D \U(D). The irreducible divisor graph of x, denoted G D (x), is given by (V, E) where the vertex set V = {Irr(x) : x D}, and given y 1, y 2 V, there is an edge {y 1, y 2 } E between vertices y 1 and y 2 if and only if y 1 y 2 x. When it is clear from context, we will drop the subscript D from G D (x) and write G(x). If the same element a Irr(D) appears multiple times in a particular factorization of x D \U(D), then we add one or more loops to the vertex a in G(x). We place n loops 18

26 CHAPTER 2. FACTORIZATION STUDIES 19 on vertex a provided a n+1 x and a n+2 x. When a vertex has more than one loop we will denote the number of loops in the graph with a superscript over the loop. Example Let D = Z[ 5] and consider the irreducible divisor graph G(18). From Example we recall that 18 factors as: 18 = = 3(1 + 5)(1 5) = 2(2 + 5)(2 5). To simplify notation we set α = (1 + 5) and β = (2 + 5), with α and β denoting their complex conjugates. Using the rules provided in Definition we construct the irreducible divisor graph shown in Figure 2.1. For example, {2, α} is an edge in G(18) since 2α 18. Since but , we place a single loop on vertex 3. We note G(18) is connected but not complete. 2 3 β β α α Figure 2.1: G(18) in Z[ 5] One of the goals in studying irreducible divisor graphs is to be able to analyze an irreducible divisor graph and to draw conclusions about the factorization of the element in question and the domain the element resides in. When we look closely at G(18) and briefly try to forget what the factorizations of 18 look like, we can see that there will be some factorization that will include β, β, and 2. Since β and β are connected by an edge in G(18), 18 = ββx for some x Z[ 5]. Similarly, 18 = 2βy and 18 = 2βz, for some y, z Z[ 5]. Since all irreducible factors of x appear together with β and β in a factorization of 18 and since

27 CHAPTER 2. FACTORIZATION STUDIES 20 none of 2, β, or β are looped in G(18), it must be the case that x = 2. Similarly, y = β and z = β. Thus 18 factors as 18 = 2ββ, and this factorization corresponds to the complete subgraph with vertex set {2, β, β}. Note that the maximal complete subgraphs {2, 3} and {3, α, α} also correspond to factorizations of 18. However, this correspondence requires a priori knowledge of the factorizations of 18 in Z[ 5] and we cannot see, simply by looking at the graph G(18) what the remaining factorizations of 18 are. This problem occurs because of the loop on the vertex 3. When we look at the graph, we really have no way of assigning the element 3 2 to any one factorization. As irreducible divisor graphs get more complicated, with more irreducible divisors, we will have much difficulty in deciphering what the factorization of a particular element is by simply looking at its irreducible divisor graph. We now look at another example. Example Let D = Z[ 5] and consider G(108). Recall from Example that 108 factors as: 108 = = (1 + 5)(1 5) = 2 2 3(2 + 5)(2 5) = 3(1 + 5) 2 (1 5) 2 = 2(1 + 5)(1 5)(2 + 5)(2 5). As before, let α = (1 + 5) and β = (2 + 5) with α and β denoting their complex conjugates. The irreducible divisor graph is given in Figure 2.2. We discuss some of the difficulties regarding this irreducible divisor graph later in Example

28 CHAPTER 2. FACTORIZATION STUDIES 21 α α β β Figure 2.2: G(108) in D = Z[ 5] We now turn to several important results that can be found in [2]. The first result gives necessary and sufficient conditions for an atomic domain to be a UFD. The second result gives a bound on the elasticity of an element given by its irreducible divisor graph. Theorem [2, Theorem 2.1] Let D be an atomic domain. The following statements are equivalent. (1) D is a UFD. (2) G(x) is complete for all x D \U(D). (3) G(x) is connected for all x D \U(D). Proposition [2, Proposition 4.1] Let x be a non-irreducible element of a BFD D. Then ρ(x) 1 max {t + l : G(x) contains a complete subgraph with t vertices and l loops}. 2 This result will be improved upon in Proposition In the special case that x is square free we produce a more accurate result.

29 CHAPTER 2. FACTORIZATION STUDIES 22 Corollary [2, Corollary 4.4] Let x be a non-irreducible element of a domain D such that a 2 x for any irreducible a in D. Then ρ(x) 1 max{t : G(x)contains a complete subgraph with t vertices}. 2 We also improve upon this result in Proposition

30 Chapter 3 The Irreducible Divisor Simplicial Complex 3.1 Introduction As promised, we now extend the concept of irreducible divisor graphs to higher dimensions. We do this in the hopes that this extension will yield more information about the factorization of elements in an atomic domain. Definition Let D be an atomic domain and let x D \U(D). The irreducible divisor simplicial complex of x, denoted S D (x), is given by (V, F ) with vertex set V given by V = {Irr(x) : x D} and with {y 1, y 2,..., y n } F if and only if y 1 y 2 y n x. In addition, to satisfy Definition , we also put F. Whenever the context is clear we will drop the subscript D from S D (x) giving S(x). 23

31 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 24 Remark Irreducible simplicial complexes are simplicial complexes. Proof. Let S(x) be an irreducible simplicial complex given by (V, F ) where V = {y Irr(x) : y x} and {y 1, y 2,..., y n } F if and only if y 1 y 2 y n x. First, we note that for all y Irr(x), {y} F since y x. Second, suppose that σ F and τ σ. Since σ F we know σ = {y 1,..., y n } where y 1 y n x. Hence τ = {y i1,..., y ij }, some sub-collection of the y i, and clearly y i1 y ij x. Thus, τ F. We graphically represent irreducible divisor simplicial complexes and irreducible divisor graphs in similar ways. Points represent vertices and edges represent faces of dimension 1. If we have some element x which factors into irreducibles as x m x mn n with distinct irreducible x i and m i 1 for all i, then the vertex representing x i will be drawn with m i 1 loops. Graphically we illustrate 2-dimensional faces by shaded triangles and 3-dimensional faces by solid tetrahedra. Again, we have no effective way to graphically depict higher dimensional faces. Example Recall the irreducible divisor graph G(18) in Figure 2.1. We now show the corresponding irreducible divisor simplicial complex S(18). 2 3 β β α α Figure 3.1: S(18) in Z[ 5] Here we have the same general structure as G(18), but we now have two dimensional facets {2, β, β} and {3, α, α} which are represented graphically as shaded faces. We avoid the

32 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 25 difficulty in determining factorizations as in Example Indeed, the facets {2, 3}, {2, β, β} and {3, α, α} correspond directly to the factorizations of 18. We will make this idea more precise in Propositions and Example We now consider the irreducible divisor simplicial complex S(108) in D = Z[ 5] Recall from Example that 108 factors as: 108 = = αα = 2 2 3ββ = 3α 2 α 2 = 2ααββ. If we investigate Figure 2.2 we can see the difficulty in extracting a particular factorization by simply analyzing the graph. However, this becomes much easer if we consider the irreducible divisor simplicial complex S(108) = (V, F ), where V = {{2}, {3}, {α}, {α}, {β}, {β}}, and F = V F 1 F 2 F 3 F 4, where F 1 = {{2, 3}, {2, α}, {2, α}, {2, β}, {2, β}, {2, γ}, {2, γ}, {α, α}, {3, α}, {3, α}, {3, β}, {3, β}, {β, β}, {α, β}, {α, β}, {α, β}, {α, β}}, F 2 = {{3, α, α}, {2, 3, α}, {2, 3, α}, {2, α, α}, {2, 3, β}, {2, 3, β}, {2, β, β}, {3, β, β}, {α, β, β}, {α, β, β}, {α, α, β}, {α, α, β}, {2, α, β}, {2, α, β}, {2, α, β}, {2, α, β}, {2, α, β}}, F 3 = {{2, 3, β, β}, {2, 3, α, α}, {α, α, β, β}, {2, α, β, β}, {2, α, β, β}, {2, α, α, β}, {2, α, α, β}}, and F 4 = {2, α, α, β, β}. The maximal faces (facets) of S(x) are {2, α, α, β, β}, {2, 3, β, β}, and {2, 3, α, α}. Unlike in G(108), we can actually see that there are factorizations of 108 that contain 2, α, α, β, and β, since they form a face of S(108). We can also conclude that there is a

33 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 26 α α 2 3 β Figure 3.2: S(108) in Z[ 5] β factorization of x that only contains 2, 3, α, and α, a fact that is not immediately apparent when examining G(108). If we consider only G(108) and consider the set A = {2, 3, α, α}, we see no clear way of proving that a factorization of 108 given by 2 3 αα will not include β or β. After all, there are edges connecting β or β to each element of A. In other words, the graph G(108) does not seem to provide enough information to support the conclusion that 108 = 2 i 3 j α k α l, for i, j, k, l Main Results In this section we provide our main results, some of which we have alluded to in the earlier examples. The first result shows that the irreducible divisor graph of an element is a subsimplex of the irreducible divisor simplicial complex of that element.

34 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 27 Proposition Let D be an atomic domain and let x D \U(D). Then K 1 (S(x)) = G(x). Proof. Let G(x) = (V, E) denote the irreducible divisor graph of x, and let S(x) = (V, F ) denote the irreducible divisor simplicial complex of x. By definition, V = V = Irr(x). Furthermore, E F since, if {a, b} E, then ab x and hence {a, b} F. Moreover, if {a, b} is an 1-dimensional face of F, then ab x and hence {a, b} E. That is, the 1-dimensional faces of S(x) are precisely the edges of G(x). S D (x). The following results give a means for finding factorizations of an element x by considering Proposition Let D be an atomic domain and let x D \U(D). Let A = {a 1,..., a n } be a facet of the irreducible divisor simplicial complex S(x). Then there exists a factorization of x given by x = a m 1 1 a m 2 2, where m i 1 for each i. Proof. Since A is a face of S(x) we know that a 1 a n x. Suppose, by way of contradiction, that there exists some factorization of x of the form a m 1 1 a mn n b 1 b k where each b j is irreducible and b j is not an associate of a i for all i, j. Then, by the definition of S(x), {a 1,..., a n, b 1 } is a face of S(x) properly containing A, contradicting the fact that A is a facet of S(x). The converse to Proposition does not hold in general as seen in Example Indeed, 108 = = 3α 2 α 2 and yet neither {2, 3} nor {3, α, α} is a facet since they are properly contained in the facet {2, 3, α, α}.

35 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 28 However, if we apply an additional restriction, we find a partial converse. Proposition Let D be an atomic domain and suppose x D \U(D) is square free. Then every factorization of x corresponds to a facet of S(x). Proof. By way of contradiction, suppose there exists a factorization x = a 1 a 2 a n, with each a i irreducible, corresponding to the face A = {a 1, a 2,..., a n } of S(x) that is not a facet. That is, A B for some facet B = {a 1, a 2,..., a n, b 1, b 2,..., b m } of S(x), where no b i is associate to any a i. Applying Proposition and the fact that x is square free, the facet B corresponds to the factorization x = a 1 a 2 a n b 1 b 2 a m. Setting these two factorizations equal we have: x = a 1 a 2 a n = a 1 a 2 a n b 1 b 2 b m. Canceling yields 1 = b 1 b m, a contradiction since each of the b i is a non-unit irreducible of D. Hence A = B is a facet of S(x). We now produce a result of great value by providing another necessary and sufficient condition for an integral domain D to be a UFD. Theorem Let D be an atomic domain. The following are equivalent: 1. For every x D \U(D), S(x) = P(A) for some A Irr(x). 2. D is a UFD. Proof. Assume 1 and let x D \U(D). Then S(x) = P(A) for some A Irr(x). Since G(x) = K 1 (S(x)) by Proposition and since K 1 (P(A)) is a complete graph, G(x) is complete. Since this holds for all x D \U(D), D is a UFD by Theorem If D is a UFD any x factors uniquely as x = a m 1 1 a mn, m i 1. Then for any subset {a i1,..., a it } {a 1,..., a n }, a i1 a it x and hence F (S(x)) = P({a i,..., a n }). That is, S(x) = P(Irr(x)). n

36 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 29 We now examine another necessary and sufficient condition for an integral domain D to be a UFD. First we provide a useful Lemma. Lemma Let a, b D \U(D) where D is a UFD. Then V (S(ab)) = V (S(a)) V (S(b)). Proof. Let x V (S(ab)). Then x ab with x irreducible and hence prime. If x a, then x V (S(a)). If x a, then x b and hence x V (S(b)). Thus x V (S(a)) V (S(b)). Conversely, suppose x V (S(a)) V (S(b)). If x V (S(a)), then x a. If x V (S(b)), then x b. In either case, x ab and hence x V (S(ab)). Remark Note that the containment V (S(a)) V (S(b)) V (S(ab)) does not require that D be a UFD. Theorem Let D be an atomic domain. The following are equivalent 1. S(a) S(b) = S(ab) for all a, b D \U(D). 2. D is a UFD. Proof. Suppose D is not a UFD. Then there exists an irreducible z D that is not prime. That is, there exists a, b D where z ab, but z a and z b. Since z ab, z V (S(ab)). We now consider S(a) S(b). By definition z V (S(a)) and z V (S(b)) and hence z V (S(a)) V (S(b)). But then z V (S(a) S(b)), since V (S(a) S(b)) = V (S(a)) V (S(b))) by Definition Therefore S(a) S(b) S(ab). Now let D be a UFD and let a, b D \U(D). We want to show that S(a) S(b) = S(ab). Since D is a UFD we know from Theorem that for any x D \U(D), S(x) = P(V (S(x))). From Lemma 3.2 we have V (S(ab)) = V (S(a)) V (S(b)). Also using Lemma ,

37 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 30 S(ab) = P(V (S(ab))) = P(V (S(a)) V (S(b))) = P(V (S(a))) P(V (S(b))) = S(a) S(b). Thus S(ab) = S(a) S(b), for all a, b D \U(D). 3.3 Elasticity We conclude this chapter by showing that irreducible divisor simplicial complexes provide bounds on the elasticity of an element that are at least as good as the results achieved in Proposition through the use of irreducible divisor graphs. Proposition Let D be a BFD. For x D \U(D) a non atom, let A(x) = {v + l : S(x) contains a facet with v vertices and l loops} and let B(x) = {v + l : G(x) contains a complete subgraph with v vertices and l loops}. Then ρ(x) 1 2 max(a(x)) 1 2 max(b(x)). Proof. Let {a 1,..., a v } be a facet of S(x) with a total of l loops on these vertices. Then a 1 a v x and thus {a 1,..., a v } is the vertex set of a complete subgraph of G(x). Loops are preserved when moving from S(x) to G(x). Therefore if n A(x), then n B(x). Thus 1 2 max(a(x)) 1 2 max(b(x)). Now we show that ρ(x) 1 max(a(x)). Since x is not 2 irreducible, min(l(x)) 2. Let M = max(l(x)). Then we can write x = a n 1 1 a n 2 2 a nt t t where the a i are distinct irreducibles and n i = M. The set {a 1, a 2,..., a t } is a face in i=1 S(x) which is contained in some facet of S(x). Also, for each i with 1 i t, there are n i 1 loops drawn on the vertex a i. Thus for any factorization of x of length M we can find a facet of S(x) that contains at least M vertices/loops. Example Consider the element 18 in the ring Z[ 5]. Recall the irreducible divisor graph G(18) in Figure 2.1. In order to maximize the total number of vertices and loops in

38 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 31 a complete subgraph of G(18) we select the complete subgraph generated by the vertices {3, α, α}. By Proposition this provides the bound ρ(18) 1 (4) = 2. If we consider 2 S(18) in Figure 3.1, we see that in order to maximize the total of the cardinality of and the number of loops in a facet of S(18) we must consider the facet {3, α, α}. Thus by Proposition ρ(18) 1 (3 + 1) = 2. In this case we see that the bound on ρ(18) is the 2 same for irreducible divisor graphs and irreducible divisor simplicial complexes. Note that by considering the factorizations of 18, ρ(18) = 1. Example Consider G(108) in Figure 2.2. In this case the graph G(108) is complete and thus we count all vertices and all loops. By Proposition ρ(108) 1 11 (6 + 5) = 2 2. Now consider S(108) in Example In order to maximize the total of vertices of and loops in a facet of S(108) we select the facet {2, α, α, β, β}. By Proposition ρ(108) 1 (5 + 3) = 4. In this case the elasticity bound as determined using the irreducible divisor 2 simplicial complex provides a better result. Note that by considering the factorizations of 108, ρ(108) = 1 Example Now consider the element x 17 in the ring D = F[x 4, x 5, x 6, x 7 ]. Factoring x 17 yields x 17 = x 4 x 6 x 7 = (x 4 ) 3 x 5 = (x 5 ) 2 x 7 = x 5 (x 6 ) 2. From this factorization we create the irreducible divisor graph G(x 17 ) and simplicial complex S(x 17 ) shown in Figures 3.3 and 3.4 respectively. In this case the graph G(x 17 ) is complete and thus we count all vertices and all loops. By Proposition we have the weaker bound ρ(x 17 ) 1 (4 + 4) = 4. 2 In order to maximize the total of vertices of and loops in a facet of S(x 17 ) we select the facet given by vertices {x 4, x 6, x 7 }. By Proposition ρ(x 17 ) 1 (3 + 3) = 3. Here we have shown that the irreducible divisor simplicial complex 2 yields a more accurate bound than the irreducible divisor graph. Note that by considering the factorizations of x 17, ρ(x 17 ) = 4 3

39 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX 32 2 x 4 x 5 x 6 x 7 Figure 3.3: G(x 17 ) in F[x 4, x 5, x 6, x 7 ] 2 x 4 x 5 x 6 x 7 Figure 3.4: S(x 17 ) in F[x 4, x 5, x 6, x 7 ] In the special case that x is square free we can determine the elasticity precisely when using irreducible divisor simplicial complexes, which is a vast improvement over the bound given in Proposition Proposition Let D be a BFD, let x D \U(D) be square free. Suppose β F (S(x)) with maximal cardinality. and α F (S(x)) with minimal cardinality. Then ρ(x) = dim(β) + 1 dim(α) + 1. Proof. By definition, ρ(x) = max(l(x)), and by Proposition each factorization of min(l(x)) x corresponds to a facet of S(x). Clearly max(l(x)) = max{ β : β F (S(x))} and min(l(x)) = min{ α : α F (S(x))}. Therefore ρ(x) = dim(β) + 1 dim(α) + 1.

40 CHAPTER 3. THE IRREDUCIBLE DIVISOR SIMPLICIAL COMPLEX Conclusion We conclude this thesis by offering topics for future study related to irreducible divisor simplicial complexes. First, irreducible divisor simplicial complexes may offer a promising way to study other commonly used invariants related to factorization such as the catenary or the tame degree of an element. Also, it may be possible to obtain tighter bounds on the elasticity of an element than we have so far achieved. In [2] the idea of a compressed irreducible divisor graph is studied. This is a graph similar to G D (x) but where some collections of vertices are compressed into a single vertex if they always occur together in a factorization of x. Can a similar compression be applied to irreducible divisor simplicial complexes? If so, what additional insight would be gained through this construction? Could we extend the notion of irreducible divisor simplicial complexes to monoids and rings with zero divisors as studied in [3]? Finally, we ask a rather large open question. Most of our examples centered around the irreducible divisor graph of a single element in a domain. What can irreducible divisor simplicial complexes tell us about the domain itself?

41 Bibliography [1] M. Axtell and J. Stickles, Irreducible divisor graphs in commutative rings with zero divisors, Comm. Alg, 36, no , [2] M. Axtell, N. Baeth, J. Stickles, Irreducible divisor graphs and factorization properties of domains, Comm. Alg, 39, no , [3] M. Axtell and J. Stickles, Irreducible divisor graphs in commutative rings with zero divisors, Comm. Alg, 36 no. 5, , [4] D. Bachman, N. Baeth, C. Edwards, Irreducible Divisor Graphs for numerical monoids, to appear in INVOLVE, A Journal of Mathematics. [5] J. Coykendall and J. Maney, Irreducible Divisor Graphs, Comm. Alg, 35, , [6] A. Geroldinger and F. Halter-Koch, Non-unique factorizations: algebraic, combinatorial, and analytic theory, Pure and Applied Mathematics. Chapman & Hall/CRC, [7] J. Maney, Irreducible divisor graphs II, Comm. Alg, 36 no. 9, , [8] H. Smallwood and D. Swartz, An investigation of the structure of underlying Irreducible Divisors, American Journal of Undergraduate Research,36 no. 2&3, 5-12, [9] H. Smallwood and D. Swartz, Properties of the diameter and girth of the hybrid irreducible divisor graph, preprint. 34