ON UPPER TRIANGULAR TROPICAL MATRIX SEMIGROUPS, TROPICAL MATRIX IDENTITIES AND T-MODULES

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1 ON UPPER TRIANGULAR TROPICAL MATRIX SEMIGROUPS, TROPICAL MATRIX IDENTITIES AND T-MODULES A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2017 By Matthew Taylor School of Mathematics

2 Contents Abstract 4 Declaration 5 Copyright 6 Acknowledgements 7 Notation 8 1 Introduction 10 2 Upper Triangular Tropical Matrices Preliminaries Green s Relations Deficiencies and Regularity Maximal Subgroups Tropical Matrix Semigroup Identities The Bicyclic Monoid Identities for U n (FT) Towards Identities for M n (T) T-modules and their Properties Elementary T-module Properties Congruences on T Classification of 2-generated T-modules List of 2-generated T-modules No Two Modules From The List Are Isomorphic

3 6 Classification of Congruences in T The List is Exhaustive A Proofs That are T-modules 210 B List of Congruences on T C Minor Proofs in Section Word Count:

4 Matthew Taylor University of Manchester Doctor of Philosophy On Upper Triangular Tropical Matrix Semigroups, Tropical Matrix Identities and T-Modules April 17, 2017 Abstract: In this thesis, we study a number of problems either directly related to or linked with tropical matrices. We shall begin by exploring the algebraic structure of finitary upper triangular tropical matrices, determining Green s relations, a combinatorial idempotency condition and maximal subgroups. We continue by proving a family of identities satisfied by these matrices and illustrating a mistake in the literature regarding weak-permutability of semigroups of tropical matrices. In the second half, we consider finitely-generated T-modules, some of which arise as solution sets to equations of the form Ax Bx, where A, B are tropical matrices. After developing several structure theorems and a powerful isomorphism invariant, we use a novel congruence-theoretic approach to classify all 2-generated T-modules, many of which do not embed in T n for any n. 4

5 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 5

6 Copyright i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see uk/docuinfo.aspx?docid=24420), in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations (see and in The University s policy on Presentation of Theses 6

7 Acknowledgements To mangle a phrase of Tolkein s, I have half the space to thank half of you half as much as I d like. First and foremost, I d like to thank my supervisor, Mark Kambites. His gentle guidance and endless patience are two huge reasons this thesis exists, and had I been somebody else s student I may not even have made it this far. Thanks to all those in the university - too many to list - who I ve had the honour of working with, sharing an office with, or simply spending my lunchtimes with. Thanks also to those who have made my life outside it, both in Manchester and London; to the writers, Magic players, pub quiz team, LSC members, Math Class gamers and Camden crowd who made so many of my evenings so special. I ll never forget a single one of you. There are a few others I d like to give special thanks to: to Jamie, for being there for a drink at Molly s when I called; to Tash, for staying on Skype until the small hours when I couldn t sleep; to Katy, for encouraging me to find help when I needed it; to David, for the many long evenings on Mumble and in Redhill; and to Andy and Dan, for friendships as long as I can remember. I could spend a whole page thanking each of you and it wouldn t be enough. No amount of pages, however, is enough to thank my parents, Clare and Phil, and my brother Mark. It s been a long road and I haven t been the perfect son or brother, but you ve always let me forge my own path - even when the choices I made surprised you. You ve supported me through thick and thin, and done more for me than I could ever possibly deserve or repay. I love you all more than I can ever say, and this thesis is for you. 7

8 Notation Semigroup Theory Notation S Usually a semigroup S 1 Semigroup S with an identity element attached E(S) The set of all idempotents in S L, R, J Green s preorders L, R, H, D, J Green s relations H e A + A The H-class corresponding to an idempotent e The free semigroup generated by an alphabet A with no empty word A + together with the empty word, the free monoid on A Tropical Matrix Notation T FT T M n (T) M n (FT) U n (T) U n (FT) R(M), C(M) GL n (T) Def M (γ) The tropical semiring, with operations and The tropical semiring without a element The completed tropical semiring, T { } All tropical matrices Tropical matrices with entries from FT Upper triangular tropical matrices Elements of U n (T) where all entries on and above the diagonal are finite The row and column space of a matrix M The invertible matrices in M n (T) The deficiency of a path γ in a matrix M 8

9 Tropical Matrix Identity Notation B The bicyclic monoid B R The real extension of the bicyclic monoid w (C,[n 1],n 1) A general (n 1)-power word of C and [n 1] (see Definition 3.2.7) G X G X,Y The digraph associated to a matrix X The compound digraph associated to matrices X and Y M, N Formal symbols corresponding to matrices (Chapter 3) C(M ij ) An operator that returns M pos(γ, k) The position in a path γ of the k th non-loop edge of γ H W (γ) The H-operator (see Definition ) T-module Notation x, y Usually elements of T n A T-module congruence (usually on T n ) X, Y Sets involved in the construction of the x-bound and y-bound (see Definition 4.2.4) k x, k y The x-bound and y-bound (see Definition 4.2.6) S The set of finitary isolated points of (see Definition ) S i, i [1, n] The shadow sets for a congruence (see Definition ) A x, A y FT { } and { } FT respectively 9

10 Chapter 1 Introduction Semigroups are one of the simplest kinds of algebraic structures imaginable - consisting of only a set together with an associative operation - and yet have had such a myriad of pages of books and papers devoted to them in their brief existence that they have already developed a sort of immortality. Though it is believed the word semigroup had been used earlier, the study of semigroups began in earnest with Anton Suschkewitsch s 1928 paper [55]. One of the first truly seminal works is considered to be J. A. Green s 1951 paper [18], which introduced the now fundamental concept of Green s relations. Let S be a semigroup, let x, y S and denote by S 1 the semigroup S with an identity element attached if necessary. Then we may define three preorders L, R, J as follows: x L y S 1 x S 1 y a S 1 : x = ay x R y xs 1 ys 1 a S 1 : x = ya x J y S 1 xs 1 S 1 ys 1 a, b S 1 : x = ayb. These preorders give rise to five equivalence relations L, R, H, D and J, known as Green s relations and defined to be: xly x L y L x xry x R y R x xhy xly and xry 10

11 11 xdy z S : xlzry xj y x J y J x. In a group, these relations are all trivial, since every element has an inverse and so we can navigate from each element to each other element. In a semigroup, however, we do not have this flexibility and so the principal ideal structure embodied in Green s relations encodes a huge proportion of the important information about the semigroup. The object S 1 is a natural example of a monoid, a semigroup with a two-sided identity element. An element e S is said to be idempotent if e 2 = e; the set of all idempotents of S is denoted E(S). Any H-class containing an idempotent is a group and, in fact, is a maximal subgroup of the semigroup; the idempotent it contains is unique, and we denote the H-class by H e (where e is the idempotent it contains). Furthermore, every maximal subgroup of S appears as the H-class of an idempotent. Green s relations and the maximal subgroup structure are two immensely important problems one should always tackle upon studying a new semigroup. Green s relations and the maximal subgroups can, to an extent, be represented pictorially by what is popularly known as an eggbox diagram. Green [18], citing Dubreil [14], observes that RL = LR as relations. This means the D-relation naturally lends itself to a representation as a grid, where the rows are L-classes and the columns R-classes, like so: An eggbox diagram, with some example idempotents marked. The individual boxes in the diagram (where L and R-classes intersect) are then H-classes. These sorts of diagrams are excellent for getting an idea of the flavour of a particular semigroup and, while not rigorous, are frequently helpful. If a semigroup only has one J -class, we call it simple; if it only has a single D-class we call it bisimple - the names derive in part from the fact that D J.

12 12 CHAPTER 1. INTRODUCTION Semigroup theory developed quickly from the 1960s onwards, with Clifford and Preston publishing the highly influential [7], split into two volumes six years apart. The subject rapidly diverged into more branches than it is possible to cover in a single survey or book: for more recent books on topics relevant to this thesis, the reader is referred to [19] or [22]; for something more recent and complex, [48]. In particular, we draw attention to the notion of inverse semigroups. We call an element a S von Neumann regular - or just regular - if there exists x S such that axa = a; we say S is regular if every element of S satisfies this property. If, furthermore, for each element a S there exists a unique x S with axa = a and xax = x, we say S is an inverse semigroup. The D-classes of these objects have particularly simple structure, as we shall see in Chapter 2. For more information, the reader should consult [35]. Another avenue into which semigroup theory turned was that of semigroup identities. The popular (and the most elegant) definition of a semigroup identity is as follows: let X = {x i : i N} denote an alphabet and X + the free semigroup generated by X. Let u, v X + be two different words. We say a semigroup S satisfies the identity u = v if, for every morphism φ : X + S, we have φ(u) = φ(v). This definition is folklore, and equivalent to saying that for every assignment of elements of S to letters x i, the resulting products in the semigroup are equal. The theory of semigroup identities is closely linked to the study of semigroup varieties. The first major paper on varieties in general is considered to be [4] by Garrett Birkhoff in although he never uses the name variety, choosing instead to refer to them as sets of algebras. What is now known as Birkhoff s Theorem (see Theorems 6 and 7, [4]) says that any class of semigroups closed under subsemigroups, direct products and morphic images can be expressed as the class of semigroups satisfying a particular collection of identities, and vice versa. In modern-day terms, this means we have two equivalent definitions of a semigroup variety: a class of all semigroups satisfying a given collection of identities; or a class of semigroups closed under subsemigroups, direct products and morphic images. While there is too much material on semigroup varieties to cover even, perhaps, in an entire thesis, the interested reader is referred to the outstanding survey articles [34] and [56]. We choose to make mention of two lines of research which are particularly relevant to the remainder of the text: firstly, Adian s 1966 work

13 13 [1], which proved what is now known as Adian s identity for the bicyclic monoid (see Chapter 3 for an explanation); secondly, the work by Lev Shneerson over several decades (a good summary of which can be found in [52]). In particular, Shneerson showed in 1993 in [51] that there exists a semigroup S such that every finitely-generated subsemigroup of S has polynomial growth but S does not satisfy a semigroup identity - resolving a long-debated conjecture analogous to the result for groups originally shown by Gromov in [20]. As it is possible to build rings from groups, one can go further with semigroups and build a semiring. Definitions vary, but for our purposes a semiring comprises a set R together with two operations + and (referred to as addition and multiplication) such that: (R, +) is a commutative monoid under addition (with identity element denoted 0); (R, ) is a monoid (with multiplicative identity denoted 1); and, for all a, b, c R, the following axioms hold: 1. a (b + c) = (a b) + (a c) 2. (a + b) c = (a c) + (b c) 3. 0 a = a 0 = 0. We note that the definitions of semiring in some texts relax the requirement of R being a monoid under multiplication or specify that the multiplication is also commutative. An idempotent semiring is one with idempotent addition operation (x + x = x for all x R). If the multiplication for a semiring admits inverses for all non-zero elements then it is called a semifield. These definitions give rise to idempotent semifields, of which we are about to see perhaps the most important example. Tropical mathematics is the study of the tropical semiring T, given by the set R { } together with the addition operation a b = max{a, b} and the multiplication operation a b = a + b. Some authors, particularly algebraic geometers, choose to use the equivalent semiring which replaces with and max with min. The tropical semiring is an idempotent semifield; more generally, it is an important example of a bipotent structure, in which a b {a, b}. Like many useful notions, the tropical semiring has lived in relative obscurity for much of its existence, but has a rich history, being studied under different names and by a broad range of academics in seemingly unrelated disciplines. Many variants also exist - one such object is T = (T,, ), with the convention

14 14 CHAPTER 1. INTRODUCTION that = ; this is known as the completed tropical semiring and we shall see it periodically in this thesis. It is only in the last ten to fifteen years, since the exploration of tropical geometry ballooned, that interest in the subject has taken off. The tropical semiring was discovered independently during the early 1970s in unpublished work by Ray Cuningham-Greene and during the late 1970s by the Hungarian Imre Simon in [53], though Simon only considered the union of the natural numbers and the element, using the min convention. The name is due indirectly to the latter; the adjective tropical was used by French mathematicians in reference to Brazil, in which Simon spent most of his career. Cuningham-Greene worked on the topic, calling it minimax algebra, through the 70s, eventually publishing the seminal book [9] - though some consider his 1962 paper [10] to be the true genesis of the subject. His work gave the tropical semiring an early home in networks, scheduling and the theory of discrete event systems - see, for example, [2] for an excellent coverage of the subjects. A more recent and equally impressive work which bridges the gap between these systems and some of the algebra behind tropical matrices is [5]. In the early 1980s, algebraic geometers began to turn their attention to the tropical semiring - Bieri and Groves [3] proved a structure theorem for tropical varieties, which related them to combinatorial objects called polyhedral complexes and would later form the basis of much of tropical geometry. The conversion of complicated algebraic varieties to heavily combinatorial objects is reflective of a general theme throughout tropical mathematics and one that will reappear in this text. Tropical geometry continued to flourish throughout the early 2000s, with two key papers appearing in 2003/04. Develin and Sturmfels developed the theory of convex polytopes in the tropical setting in [13]; Mikhalkin then showed it was possible to use tropical geometry to count the number of curves of any genus on toric surfaces in [42]. Mikhalkin s paper (eventually published in 2005) opened many algebraic geometers minds to the possibilities of tropical mathematics and is arguably the catalyst for modern tropical geometry. For further information on the ideas behind the subject, the reader is referred to the survey articles [38], [43] and the more recent book [39]. One other recent and major development which we shall discuss further in Chapter 4 is the move towards a tropical theory of schemes [16].

15 15 As with matrices over a field, one can define matrices over a general semiring S (with operations +, ) by considering the entries to be elements of S, with the induced operations given by (A + B) ij = A ij + B ij (A B) ij = A ik B kj. Of the two operations above, we shall concentrate on matrix multiplication throughout this thesis and denote by M n (T) the semigroup of all square matrices of size n over the tropical semiring together with the appropriate matrix multiplication operation. We shall refer to any matrix over the tropical semiring as a tropical matrix. We may also scale tropical matrices of any size (not necessarily square) by λ T by, given a matrix A, setting (λ A) ij = λ A ij. This frequently manifests itself in the scaling of elements of tropical n-space T n later on in the thesis. Very occasionally when dealing with T n we shall also want to consider what is commonly known as projective tropical n-space, denoted P T n 1. Intuitively, projective tropical n-space can be thought of as projecting onto a particular coordinate by scaling every element of T n appropriately. Explicitly, we have P T n 1 := T n/ =, where = is the T-module congruence which identifies two elements x, y T n if and only if x = λ y for some λ FT. Despite some isolated study of tropical matrices through the 90s (see, for example, [54] for an early consideration of their semigroup theory or [46] for a few interesting results) there had been no concerted effort to explain the algebraic structure of M n (T) until recently. Kambites, Hollings, Johnson and Izhakian have shown a series of results in [21], [25], [26], [30], [33] and [31] describing concepts such as Green s relations and von Neumann regularity via geometric arguments. The following text is split into five chapters. In Chapter 2, we consider the upper triangular tropical matrices, where M ij = for all i > j; this set, together with the matrix multiplication operation defined previously, forms a semigroup, U n (T). The set of matrices in U n (T) where all entries on and above the diagonal are finite forms a subsemigroup of U n (T), which we denote U n (FT). We demonstrate that Green s relations for U n (FT) are the restrictions of the corresponding relations for M n (T) and that this subsemigroup has exceptional combinatorial features; for example, there is only one J -class and the idempotency

16 16 CHAPTER 1. INTRODUCTION condition and D-relation can be phrased in terms of the relative weights of paths through an associated digraph. This leads to a polynomial time algorithm to decide whether two elements are D-related. We also give results pertaining to the maximal subgroups - every maximal subgroup of U n (FT) is a copy of R, and the group-bound elements are exactly those where every diagonal entry is equal. Chapter 3 concerns semigroup identities for tropical matrices. This area of research began with the observation by Izhakian and Margolis [27] that Adian s identity holds for U 2 (T) and continued more recently with further work by Okninski [45] and Izhakian [23]. We prove an adjustment of the main (incorrect) claim in [23] which gives large families of identities satisfied by U n (FT), and resolve a conjecture in the paper concerning the length of the shortest possible semigroup identity for each n. We also use embeddings of finite semigroups into M n (FT) to show that no one semigroup identity is satisfied by M n (T) for all n, and illustrate a mistake in [11] which voids a proof of weak-permutability of tropical matrix semigroups. As with defining matrices over a semiring, one can also define a module over a semiring, which we shall refer to as a semimodule (or semimodule over a semiring, or just module for short) throughout. Matrices over idempotent semifields are intrinsically linked with semimodules over those semifields. In Chapter 4 we begin an exploration of T-modules, with applications to the tropicalisation of schemes. We prove a collection of connected results concerning T-module congruences on T 2, together with some general abstract algebraic theorems which shall form the basis for the final chapters. In particular, we demonstrate a decomposition of any T-module congruence on T 2 into a max-plus convex set of what we call finitary isolated points, together with information about elementary topological properties of the remaining equivalence classes. In Chapter 5 and Chapter 6 we begin to leverage the techniques we have developed to give a classification of all 2-generated T-modules. No such list currently exists, and it would form an invaluable library of examples in which to check conjectural results about more complicated structures; there is a surprising amount of variety even in the 2-generated case, with many of the semimodules having no embedding in T n for any value of n. First, we give a list of such semimodules, which we claim is exhaustive, and show that no two semimodules in the list are isomorphic - primarily by use of an isomorphism invariant which takes the form of a directed, labelled graph called the section graph. Second, we use

17 17 the results from Chapter 4 to classify all possible semimodule congruences on T 2, enabling us to justify the claim that the previous list of 2-generated T-modules is exhaustive.

18 Chapter 2 Upper Triangular Tropical Matrices In this chapter, we shall explore the structural properties of certain upper triangular tropical matrix semigroups - specifically, we shall concentrate mainly on the case where all the entries on and above the diagonal are finite. The chapter begins with some preliminary results from existing papers and studies of analogous objects. Section 2.2 will cover Green s relations; in particular, we will examine how the relations compare to those for M n (T), which have been well-documented in [21], [31] and [30]. Section 2.3 begins by completely classifying the idempotent matrices in U n (FT) and presents a quick algorithm for determining whether a matrix is regular (which, in turn, extends naturally to an algorithm to determine whether two elements are D-related). The section finishes with a geometric link to the J -relation, which suggests a particularly beautiful result. Finally, Section 2.4 classifies all maximal subgroups of U n (FT) and discusses group-bound elements. 2.1 Preliminaries Before continuing with the main body of the thesis, we present some preliminary definitions and results to put our work into context. The first of these definitions is folklore in tropical mathematics. Definition Let S T n. We say S is a max-plus convex set if it is closed under the maximum operation and scaling. 18

19 2.1. PRELIMINARIES 19 Remark We allow S to be empty, though in the remainder of the thesis we shall almost exclusively consider non-empty max-plus convex sets. Note that this is a different concept to that of a subsemiring, as there is no requirement that the componentwise operation on T 2 keeps us in the max-plus convex set. A max-plus convex set does, however, form a subsemimodule of T n. We begin in earnest by recapping a few definitions and an important result from [6]. Though Butkovic, Schneider and Sergeev chose in that paper to work with the semiring (R 0,, ), where denotes regular multiplication, we note this semiring is isomorphic to our definition of T. They also use the term max cone in place of max-plus convex set. Definition ([6], Definition 2). A max combination of S T n is any vector of the form v = λ x x, x S where λ x T for all x and only finitely many λ x are not equal to. For our purposes, we will usually consider max combinations of a finite generating set, so the final condition in the definition will be irrelevant; observe also that the set of all max-plus combinations of a set S T n is a max-plus convex set. Definition ([6], Definition 3). Let S T n. An element x S is an extremal of S if, for any u, v S such that x = u v, we have either x = u or x = v. Remark If n = 1 in the above definition, so S T, then all elements of S are extremal. The following is a paraphrasing of Definition 4, from [6]. Definition ([6], Definition 4). An element x T n is scaled if the maximum 1 i n x i = 0. We have chosen 0 as our scaled value, where [6] used 1. This is a reflection of our choice of operations and underlying set - observe that 1 is the multiplicative identity in the semiring (R 0,, ) and that 0 is the multiplicative identity for T.

20 20 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES The authors phrase their results in [6] in terms of scaled elements and scaled extremals, in order that they may use the terminology unique scaled basis for a max-plus convex set. In reality, there are often many minimal generating sets for a max-plus convex set, since by scaling any of the generators by a finite constant we obtain a new minimal generating set. The following is a shorter version of Theorem 18 from [6], phrased using our own terminology. There is a third equivalent statement, but it is not required in the body of the thesis. Theorem ([6], Theorem 18). Let E be the set of scaled extremals of a max-plus convex set K. Let S K be a set of scaled elements. The following are equivalent: 1. S is a minimal generating set for K; 2. S = E and S is a generating set for K. Remark In particular, it follows immediately that the minimal generating sets for K are exactly the sets of the form {λ 1 x 1, λ 2 x 2,..., λ n x n }, where { x 1, x 2,..., x n } = E and λ 1, λ 2,..., λ n FT. Green s relations for M n (T) are well-understood; we continue by recapping the existing results on the topic. Definition Let M M n (T) be a tropical matrix. The row space of M, denoted R(M), is the set of all max combinations of rows of M, so that R(M) T n ; the column space, denoted C(M), is the set of all max combinations of columns of M. The concepts of row and column space for tropical matrices are closely related to their algebraic structure under matrix multiplication. The following are the relevant results of Hollings, Kambites and Johnson, abbreviated for our purposes where necessary. Proposition ([30], Corollary 3.2). Let M, N M n (T). Then MLN in M n (T) (respectively, MRN in M n (T)) exactly if R(A) = R(B) (respectively, C(A) = C(B)) in affine tropical n-space. Remark This result has been extended by Kambites and Hollings ([21], Proposition 4.1) to one where T is replaced by any commutative semiring with

21 2.1. PRELIMINARIES 21 a multiplicative identity and local zeros. In particular, the same result is true when T is replaced by FT or T. Theorem ([21], Theorem 6.5). Let M, N M n (T). Then the following are equivalent: 1. MDN in M n (T); 2. R(M) and R(N) are algebraically isomorphic (as T-modules); 3. C(M) and C(N) are algebraically isomorphic (as T-modules). In the final theorem concerning Green s relations, if M M n (T) is a matrix over the completed tropical semiring then R T (M) is the T-linear analogue of the row space, where we allow scaling by in our max-combinations; C T (M) is the column space analogue. Theorem ([31], Theorem 5.3). Let M, N M n (T). Then the following are equivalent: 1. M J N in M n (T); 2. there exists a max-plus convex set Y T n such that R T (M) Y and there is a surjective T-module morphism φ : R T (N) Y ; 3. there exists a max-plus convex set Y T n such that C T (M) Y and there is a surjective T-module morphism φ : C T (N) Y. Though the result for the J -relation is in terms of matrices over T, Proposition 4.1 of [31] says that the J -relation is inherited; that is, two matrices in M n (T) are J -related in M n (T) if and only if they are J -related in M n (T). Therefore the above result is also true for the ordinary tropical matrices we will be working with. Many of our arguments in Chapter 2 use particular types of rank of tropical matrices. The following is a recap of existing, well-known definitions - there are many concepts of rank for a tropical matrix (see [31] for an excellent summary) but we shall focus mostly on the row and column ranks. Definition The row rank of M is equal to the size of a minimal generating set for R(M) in T n ; this value is also known as the generator dimension of R(M). The column rank of M is equal to the size of a minimal generating set for C(M) in T n ; this value is also referred to as the dual dimension of R(M).

22 22 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES In their paper on the J -relation, Johnson and Kambites also showed that these concepts of rank were D-class invariants in M n (T). Proposition ([31], Corollary 8.5). Let M, N M n (T) such that MDN. Then the row rank of M is equal to the row rank of N, and the column rank of M is equal to the column rank of N. We shall see in Section 2.2 that matrices in U n (FT) have row and column rank n; the above proposition will then be of paramount importance in proofs concerning the D and J -relations. 2.2 Green s Relations With the study of any semigroup, the initial questions that are asked are inevitably about Green s relations and the structure of the maximal subgroups. For example, while D = J for any finite semigroup, this is not necessarily true for general semigroups. Ideally, we would hope for Green s relations on U n (FT) and U n (T) to be the restrictions of the corresponding relations for M n (T). We begin with a very useful result which we shall use to study Green s relations on U n (FT) - one which shall frequently be used in future proofs. Proposition Every M U n (FT) has both row and column rank n. Proof. We prove M has row rank n, the case for column rank being dual. For i N with 1 i n, let r i denote the i th row of M. Assume S T n is a finite set which generates R(M) - such a set must exist, since we can take the rows of M. Let s 1, s 2,..., s m denote the elements of S and let ( s i ) j denote the j th coordinate of s i. We claim that, for all k N such that 1 k n, there exists t S such that t 1, t 2,..., t k 1 = and t k, from which the result follows. Indeed, we have for some set of λ i T, implying r k = (( r k ) 1, ( r k ) 2,..., ( r k ) n ) = ( r k ) k = 1 i m 1 i m λ i ( s i ) k. λ i s i

23 2.2. GREEN S RELATIONS 23 In particular, for each k, there must exist an element t = s a S with λ a and ( t) k. However, for all j N such that 1 j k 1, we have = ( r k ) j = λ i ( s i ) j, 1 i m implying that ( s a ) j = ( t) j = as required, completing the proof of the claim and of the proposition. Having full row and column rank is an advantageous property in M n (T) - many of the concepts in tropical mathematics (see, for a good example, [26]) require separate cases or considerable effort to prove for lower rank matrices. We require one further result to aid our manipulations later, for which we recap the concept of invertible matrices: Definition We denote by GL n (T) the group of all invertible matrices in M n (T), where here our definition of an invertible matrix is a matrix G such that there exists G 1 with GG 1 = G 1 G = E, the identity matrix. It is wellknown (see, for example, Section 7 of [26]) that GL n (T) comprises precisely the monomial matrices in M n (T) - that is, all matrices with exactly one finite entry in each row and exactly one finite entry in each column. We say a matrix in GL n (T) is diagonal if its only finite entries are those on the diagonal. Contrasting this with the case of matrices over a field, where all matrices with non-zero determinant are invertible, we see that invertibility is far from the norm in M n (T). Proposition Fix M, N U n (FT) such that MLN (respectively, MRN) in M n (T). Then there exist diagonal matrices A, B GL n (T) such that AM = N and BN = M (respectively, diagonal matrices C, D GL n (T) such that MC = N and ND = M). Proof. We prove the L-relation case, the proof for the R-relation being dual. Recall from Proposition that two matrices in M n (T) are L-related if and only if they have the same row space. Let r 1, r 2,..., r n and s 1, s 2,..., s n be the rows of M and N respectively. By Proposition 2.2.1, M and N have full row rank. Since MLN in M n (T), Proposition gives that R(M) = R(N). We claim that a scaling of each row in

24 24 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES M must appear as a row of N. Indeed, Theorem and Remark imply that the elements of any minimal generating set for R(N) must be scalings of the elements of any minimal generating set for R(M). But since M and N have full row rank, their rows each form minimal generating sets for R(M) and R(N) respectively, meaning each row of N must be a scaling of a unique row of M. Therefore, there exist σ S n and λ 1, λ 2,..., λ n FT such that s i = λ i r σ(i) for all i {1, 2,..., n}. Since r i and s i are the only rows of M and N respectively which have i finite entries, we must have s i = λ i r i. Let G GL n (T) be given by G ii = λ i for all i {1, 2,..., n} and G ij = for all i, j {1, 2,..., n} with i j. Let H GL n (T) be given by H ii = λ i for all i {1, 2,..., n} and H ij = for all i, j {1, 2,..., n} with i j. Then GM = N and HN = M. This completes the proof. The submonoid of diagonal matrices in GL n (T) is particularly useful, with a plethora of desirable properties - for example, it is commutative and there is both a natural left and right action of GL n (T) (given by tropical matrix multiplication) on U n (T). Furthermore, these actions restrict to actions on U n (FT). The following lemma begins our study of whether Green s relations for U n (FT) are the restrictions of the analogous relations in M n (T). Lemma The L and R-relations on U n (FT) are the restrictions of the corresponding relations on M n (T). Proof. We prove the lemma in the case of the L-relation, with the R-relation case being dual. Recall from Proposition that for two matrices M, N M n (T), we have MLN in M n (T) A, B M n (T) : AM = N, BN = M R(M) = R(N). If M, N U n (FT) and MLN in U n (FT), then - by definition - there exist matrices C, D U n (FT) such that CM = N, DN = M. Then we have MLN in M n (T), because we can take A = C and B = D in the above equivalence. Now assume M, N U n (FT) and MLN in M n (T). By Proposition 2.2.3, there exist diagonal matrices G, H GL n (T) such that GM = N, HN = M. Fix such a pair of matrices G, H. We define new matrices G, H U n (FT) whose finite entries are given by

25 2.2. GREEN S RELATIONS 25 G ii = G ii, H ii = H ii and for 1 i < j n, G ij = min j k n {G ii (M ik M jk )} H ij = min j k n {H ii (N ik N jk )} Then (G M) ij = 1 k n G ik M kj. However, since G, M U n (FT), we may restrict the values over which we take the maximum to those between i and j; indeed, if k < i then G ik = and if k > j then M kj =, so these terms will not contribute to the maximum. This is a technique we shall use frequently throughout the thesis. Thus, we have (G M) ij = G ik M kj = i k j = (i+1) k j (i+1) k j M kj min k l n {G ii (M il M kl )} (G ii M ij ) G ii M kj min k l n {M il M kl } (G ii M ij ). Next, note that M kj min k l n {M il M kl } M ij since M ij M kj is part of the minimum. So (G M) ij = (G ii M ij ) x ij, where x ij G ii M ij. Thus (G M) ij = G ii M ij, implying G M = GM = N. Since G U n (FT), we have N L M in U n (FT). Clearly, a symmetrical argument gives H N = HN = M, so we have MLN in U n (FT). We can deduce a similar result for the H-relation as an almost immediate corollary. Corollary The H-relation in U n (FT) is the restriction of the H-relation in M n (T).

26 26 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES Proof. By Lemma 2.2.4, two matrices M, N U n (FT) are L-related (respectively, R-related) in U n (FT) if and only if they are L-related (R-related) in M n (T). Thus: MHN in U n (FT) MLN and MRN in U n (FT) MLN and MRN in M n (T) MHN in M n (T) Given the results about restrictions of the L and R-relations, we can say that two matrices M, N U n (FT) are L-related if and only if they have the same row space (and R-related exactly if they have the same column space). Since M, N have row and column rank n - and given the connection with diagonal monomial matrices in Proposition we may therefore expect the following result: Proposition Let M, N U n (FT). Then MLN (respectively, MRN) if and only if the i th row (column) of N is a finite scaling of the i th row (column) of M. Proof. The left-to-right proof follows immediately from Lemma and Proposition For the converse, observe that if N ij = λ i M ij for all 1 i n then λ i FT, N = GM and M = G 1 N where G ii = λ i for 1 i n and G ij = whenever i j, so MLN. Similarly, if N ij = µ j M ij for all 1 j n then N = MH and M = NH 1 where H GL n (T) and H jj = µ j. Before continuing, we provide a simple description of how to move around certain L and R-classes in M n (T), which will become important when we turn our attention to the D-relation. Proposition Let M, N M n (T) be matrices with row rank n (respectively, column rank n) such that M LN (respectively, M RN). Then there exists a monomial matrix G such that GM = N (respectively, MG = N). Proof. We shall prove the result for the L-relation, since the R-relation follows by a dual argument.

27 2.2. GREEN S RELATIONS 27 Let r 1, r 2,..., r n and s 1, s 2,..., s n denote the rows of M and N respectively. Since MLN, we have R(M) = R(N). Since the row rank of M and N is n, the size of a minimal generating set for R(M) (and of such a set for R(N)) is n. Therefore, by Theorem and Remark 2.1.8, there exists some σ S n and λ 1, λ 2,..., λ n FT such that, for all i {1, 2,..., n}, s σ(i) = λ i r i. Fix such a collection of λ i and a permutation σ. Let G σ(i)i = λ i for all i {1, 2,..., n} and G ij = for all other entries. Then GM = N. Recall that two elements a, b of a semigroup S are D-related if and only if there exists an intermediate element c S such that alcrb. We begin our exploration of the D-relation by describing the intermediate elements which can appear when matrices in U n (FT) are considered to be elements of the semigroup M n (T). Lemma Let M, N U n (FT) be such that MDN in M n (T). Let X M n (T) be such that MLX and XRN in M n (T) (dually, such that XRM and XLN in M n (T)). Then any G GL n (T) such that GM = X must be diagonal (dually, any H GL n (T) such that NH = X must be diagonal). Proof. We prove the first half of the lemma, the second half following from a dual argument. Recall that L, R D as relations. Then, by Proposition , the row rank of X must be equal to the row rank of M and the column rank of X must be equal to the column rank of N. Thus X has row and column rank n. Since XRN and X, N have column rank n, by Proposition we can choose a monomial matrix H such that XH = N. Fix such a matrix H. Assume there exists some non-diagonal monomial matrix G such that GM = X. Let G ij be a finite entry of G such that i > j - such an entry must exist, as G is not diagonal and so has finite entries in positions G σ(i)i for some non-trivial σ S n (and in any non-trivial permutation, there must be a value of i which is mapped to j < i). Then X ij = G ir M rj = G ij M jj 1 r n

28 28 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES since G is monomial. Also, for any k < j, we have X ik = G ir M rk = 1 r n since when r = j (corresponding to the only finite entry in row i of G) we have M jk =. Furthermore, observe that for any l > j we have X il = G ir M rl = G ij M jl FT 1 r n since M jl FT. Now there are (n j + 1) finite entries in row i of X. Multiplying X on the right by any monomial matrix has the effect of permuting and/or scaling the columns of X. So regardless of our previous choice of H, row i of XH = N has (n i + 1) finite entries. However, since we showed above that row i of X has (n j + 1) finite entries and i > j there are more entries in row i of XH than there should be, a contradiction to N U n (FT). So our initial assumption, that G is non-diagonal, must be false. This completes the proof. We can now show that the D-relation on U n (FT) is also the restriction of the corresponding relation in M n (FT). Theorem Let M, N U n (FT). Then MDN in M n (T) exactly if MDN in U n (FT). Proof. Note first that if MDN in U n (FT) then there exists a matrix X U n (FT) such that MLXRN. Since X M n (T), MDN in M n (T). Thus we need only prove the converse to prove the theorem. Assume M, N U n (FT) and MDN in M n (T). Then there exists at least one matrix X M n (T) such that MLXRN. To prove the theorem, we must show that we can choose X U n (FT). By Proposition 2.2.1, M, N both have row and column rank n. Furthermore, by Proposition the row rank of X must be equal to the row rank of M and the column rank of X must be equal to the column rank of N. Thus X has row and column rank n. Since M and X both have row rank n, Proposition applies and we can choose a monomial matrix G such that GM = X. By Lemma 2.2.8, G must be diagonal. Therefore GM = X must also be in U n (FT), implying that there exists X U n (FT) such that MLXRN in M n (T). By Lemma 2.2.4, if M, N U n (FT)

29 2.2. GREEN S RELATIONS 29 are such that MLN (respectively, MRN) in M n (T) then MLN (respectively, MRN) in U n (FT), completing the proof. This is a very useful result for the study of U n (FT), since it means we can use the characterisations of the D-relation for M n (T) in [21] to work with U n (FT). In particular, it follows from Proposition that two finitary upper triangular matrices are D-related if and only if they have isomorphic row or column spaces. Finally, we deal with the J -relation. While we shall see later that there is a beautiful geometric intuition behind the result, the proof is necessarily lengthy and algebraic. Theorem Let M, N U n (FT). Then MJ N in M n (T) exactly if MJ N in U n (FT). Proof. If MJ N in U n (FT), then there exist matrices A, B, C, D U n (FT) such that M = ANB and N = CMD. Since A, B, C, D M n (T), we have MJ N in M n (T). Thus we need only prove the converse to prove the theorem. We will do this by proving the analogous result for the J -preorder J, from which the result follows. Assume M, N U n (FT) such that M J N in M n (T). Then there exist matrices A, B M n (T) such that M = ANB. We need to show that we can choose A, B U n (FT). We begin by showing A and B must be in U n (T) and each row of A and each column of B must contain at least one finite entry. We shall show the result for A, the result for B being dual. First, we shall show A U n (T). Assume, for a contradiction, A ij is an entry of A such that i > j and A ij FT. Then, for all r 1 j, we have (AN) ir1 = A ik N kr1 1 k n because A ij N jr1. Now (ANB) ij = M ij = and (ANB) ij = (AN) ik B kj. 1 k n So, since (AN) ir1 for all r 1 j, we must have B r1 j = for all r 1 j. Given the above, for all r 2 j we have (NB) r2 j = N r2 k B kj. 1 k n

30 30 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES Since N r2 k = for all k < r 2 and B kj = for all k such that k j (and thus for all r 2 such that k r 2 j), we have (NB) r2 j = for all r 2 j. But (ANB) jj = M jj, so since (ANB) jj = A jk (NB) kj 1 k n and (NB) kj = for all k j, we must have A jr3 for some r 3 < j. Pick such an r 3. Now we can iterate this argument to obtain a sequence of integers i > j > r 3 > r 6 >... such that A ij, A jr3, A r3 r 6,... are all finite. However, since the size of the matrix is finite, the sequence i, j, r 3, r 6,... must eventually go below 1, giving a contradiction (since a matrix entry cannot be in a row or column denoted by a negative number). So our original assumption that A ij for some i > j was false and so A U n (T). Next, we show there must be at least one finite entry in each row of A. Assume there exists i {1, 2,..., n} such that, for all j i, we have A ij =. Then, for all r i, we have (ANB) ir = A ik (NB) kr i k r = which gives a contradiction to ANB = M U n (FT). Therefore A has at least one finite entry in every row. A dual argument shows that every column of B has at least one finite entry in every column. So we may assume that A, B U n (T) such that, for all 1 i n, there exists j i and k i with A ij and B ki. Let A be the matrix whose entries are given by A ij = min j k n {(AN) ik N jk } when i j and A ij =, by A ij = A ij when i j and A ij FT, and by

31 2.2. GREEN S RELATIONS 31 otherwise. Let B be the matrix whose entries are given by B ij = min 1 k i {(NB) kj N ki } when i j and B ij =, by B ij = B ij when i j and B ij FT, and by otherwise. Now, observe that, for all i such that 1 i n, we have (AN) in = A ik N kn FT i k n since N U n (FT) (so N kn FT) and at least one value of A ik must be finite in the above maximum. Therefore we have A, B U n (T). The definitions of these matrices are a little opaque without an explanation, so we pause briefly to provide some insight. The goal when constructing A and B is to build matrices which have the same effect as A and B when multiplying on the left and right respectively, but which are in U n (FT). To do this, we take A and B and change each entry into something which is finite, but small enough that it doesn t have an impact in the matrix multiplication. This is the role of the minimum in the definition of each A ij and B ij. Now, when i j, certainly we have (A N) ij (AN) ij since A ij = A ij when A ij is finite. So A N U n (FT). Assume (A N) ij > (AN) ij. Then there must exist k such that A ik = min k r n {(AN) ir N kr } and A ik N kj > (AN) ij, so min k r n {(AN) ir N kr } N kj > (AN) ij. But, since k j, we have (AN) ij < min k r n {(AN) ir N kr } N kj (AN) ij N kj N kj = (AN) ij implying (AN) ij > (AN) ij, a contradiction. Therefore (A N) ij = (AN) ij for all i, j. A dual argument shows that (NB ) ij = (NB) ij for all i, j. Therefore A N = AN and NB = NB. In particular, A NB = ANB = M

32 32 CHAPTER 2. UPPER TRIANGULAR TROPICAL MATRICES and so M J N in U n (FT). In particular, if MJ N in M n (T) then both M J N and N J M in M n (T), implying both M J N and N J M in U n (FT) and thus MJ N in U n (FT). This completes the proof. Knowledge of Green s relations lets us describe parts of the eggbox structure - later in the chapter we will examine the maximal subgroups and idempotents, giving us a near-complete understanding of U n (FT). For now, we shall delve a little into the 2 and 3-dimensional cases. These are computationally simpler and will give us an idea of what to expect when we begin working in generality. In particular, there are some general properties of U n (FT) which degenerate to existing semigroup-theoretic concepts for U 2 (FT). We shall explore these next. Recall that a semigroup is called bisimple if it only has one D-class (and therefore, since D J as binary relations, only one J -class). Proposition The semigroup U 2 (FT) is bisimple. Proof. Given two matrices M, N U 2 (FT), we show the existence of a matrix X U 2 (FT) such that C(X) = C(N) and R(M) = R(X). Let m 1, m 2 denote the rows of M and c 1, c 2 denote the columns of N. We claim the matrix ( ) N 11 N 11 M 11 M 12 X = N 22 M 12 M 11 N 11 N 12 satisfies these properties. Let r 1, r 2 denote the rows of X and s 1, s 2 denote the columns of X. Indeed, since r 1 = (N 11 M 11 ) m 1 and r 2 = (N 22 M 12 N 11 M 11 N 12 M 22 ) m 2 and the scalars are finite in both cases, we have R(M) = R(X). Then, since c 1 = 0 s 1 and c 2 = (N 12 M 11 M 12 N 11 ) s 2, we have C(N) = C(X). Therefore MLX and XRN, implying by definition of Green s relations that MDN.

33 2.2. GREEN S RELATIONS 33 Further to this, we can explain aspects of the structure of this single D-class using the medium of inverse semigroups. Proposition The semigroup U 2 (FT) is an inverse semigroup. Proof. Fix A U 2 (FT). We will construct an inverse element for A, denoted B, and show it is unique. If B U 2 (FT), the matrix product ABA is as follows: ABA = A 11 B 11 A 11 (A 11 B 11 A 12 ) (A 11 B 12 A 22 ) (A 12 B 22 A 22 ) (2.1) A 22 B 22 A 22 Set B 11 = A 11, B 22 = A 22 and B 12 = A 12 A 11 A 22. Then ABA = B and BAB = B. To finish the proof, we need to show this choice of B is unique - that there can be no C U 2 (FT) such that C B, ACA = A and CAC = C. Since (ACA) 11 = A 11 = A 11 (A 11 C 11 ) and (ACA) 22 = A 22 = A 22 (A 22 C 22 ) we must have C 11 = A 11 and C 22 = A 22. The only remaining choice is the choice of C 12. Assume C 12 < A 12 A 11 A 22. Then A 12 > C 12 A 11 A 22, so by 2.1 (with A replaced by C and B replaced by A) we would have C 11 A 12 C 22 > C 12 A 11 A 22 C 11 C 22 = C 12 and so (CAC) 12 > C 12, a contradiction. Now assume C 12 > A 12 A 11 A 22, so by 2.1 (with B replaced by C) we would have C 12 A 11 A 22 > A 12 and so (ACA) 12 > A 12, a contradiction. So our unique choice of C 12 is A 12