Semigroup identities in the monoid of triangular tropical matrices
aa r X i v : . [ m a t h . R A ] M a y SEMIGROUP IDENTITIES IN THE MONOID OFTRIANGULAR TROPICAL MATRICES
ZUR IZHAKIAN
Abstract.
We show that the submonoid of all n ˆ n triangular tropical matrices satisfies anontrivial semigroup identity and provide a generic construction for classes of such identities.The utilization of the Fibonacci number formula gives us an upper bound on the length of these2-variable semigroup identities. Introduction
Varieties in the classical theory are customarily determined as the solutions of systems of equa-tions. The “weak” nature of semigroups, i.e., lack of inverses, forces the utilization of a differentapproach than the familiar one, in which semigroup identities simulate the role of equations inthe classical theory. These semigroup identities are at the heart of the theory of semigroup vari-eties [10], and have been intensively studied for many years. A new approach for studying thesesemigroup identities has been provided by the use of tropical algebra, as introduced in [7].Tropical algebra is carried out over the tropical semiring T : “ R Y t´8u with the operationsof maximum and summation (written in the standard algebraic way), a ` b : “ max t a, b u , ab : “ a ` sum b, serving respectively as addition and multiplication [6, 8, 9]. This semiring is an additively idem-potent semiring, i.e., a ` a “ a for every a P T , in which : “ ´8 is the zero element and : “ ˆ ˆ n ˆ n tropical matrices. These matrices essentially serve as the target forrepresenting semigroups, and thus enable representations and study of a larger range of monoidsand semigroups.In the present paper we deal with n ˆ n triangular tropical matrices, improving the machineryintroduced in [7] by bringing in the perspective of graph theory. It is well known that graphtheory, especially the theory of direct graphs, is strongly related to tropical matrices [2, 3] andprovides a powerful computational tool in tropical matrix algebra; the correspondence betweentropical matrices and weighted digraphs is intensively used for proving the results of this paper. Abackground on the interplay between digraphs and tropical matrices is given in § Date : November 9, 2018.2010
Mathematics Subject Classification.
Primary: 20M05, 20M30, 47D03; Secondary: 16R10, 14T05.
Key words and phrases.
Tropical (max-plus) matrix algebra, idempotent semirings, semigroup identities, semi-group varieties, monoid representations.
Acknowledgement.
The author thanks Glenn Merlet for the useful conversation in CIRM, Luminy, September2011.
Before approaching tropical matrices, we first address semigroup identities in general (cf. § Theorem 3.10.
A semigroup that satisfies an n -variable identity, also satisfies a refined -variableidentity with exponent set t , u . This refinement of semigroup identities assists us to deal with tropical matrices by better utilizingtheir view as digraphs.The monoid M n p T q of n ˆ n matrices over the tropical semiring plays, as one would expect,an important role both in theoretical algebraic study and in applications to combinatorics, as wellas in semigroup representations and automata. In contrast to the case of matrices over a field,we identify nontrivial semigroup identities, satisfied by the submonoid U n p T q (resp. L n p T q ) of allupper (resp. lower) n ˆ n triangular tropical matrices.To simplify the exposition we open with a certain type of matrices, i.e., diagonally equivalentmatrices, and have the following preliminary theorem: Theorem 4.8.
Any two triangular tropical matrices
X, Y P U n p T q having the same diagonalsatisfy the (nontrivial) identities: r w p C,P,n ´ q X r w p C,P,n ´ q “ r w p C,P,n ´ q Y r w p C,P,n ´ q , (1.1) where r w p C,P,n ´ q is any word having as its factors all the words of length n ´ generated by C “ t X, Y u of powers P “ t , u , such that r w p C,P,n ´ q X r w p C,P,n ´ q and r w p C,P,n ´ q Y r w p C,P,n ´ q are generated by C and powers P . (To be explained in the text below.) Using this result, basically proved by combinatorial arguments on the associated (colored)weighted digraphs of products of tropical matrices, we obtain the main result of the paper.
Theorem 4.10.
The submonoid U n p T q Ă M n p T q of upper triangular tropical matrices satisfiesthe nontrivial semigroup identities (1.1) , with X “ AB and Y “ BA , for any A, B P U n p T q . This theorem generalizes the identity of the submonoid U p T q of 2 ˆ U p T q , led in [7] toan easy proof of Adjan’s identity of the bicyclic monoid (see [1] for Adjan’s original work).The well known Fibonacci number formula provides us an easy way to compute an upper boundfor the length of the 2-variable semigroup identities discussed in this paper (cf. § Background: Tropical matrices and weighted digraphs
Recalling that T is a semiring, then in the usual way, we have the semiring M n p T q of n ˆ n matrices with entries in T , whose addition and multiplication are induced from T as in the familiarmatrix construction. The unit element I of M n p T q , is the matrix with “ “ ´8 ; the zero matrix is p q “ I . Therefore, M n p T q is also a multiplicative monoid, and in the sequel it is always referred to as a monoid. Formally,for any nonzero matrix A P M n p T q we set A : “ I . A given matrix A P M n p T q with entries a i,j is EMIGROUP IDENTITIES IN THE MONOID OF TRIANGULAR TROPICAL MATRICES 3 written as A “ p a i,j q , i, j “ , . . . , n . We denote by U n p T q (resp. L n p T q ) the submonoid of M n p T q of all upper (resp. lower) triangular tropical matrices.Given two matrices X “ p x i,j q and Y “ p y i,j q in M n p T q , we write X „ diag Y ô x i,i “ y i,i , for all i “ , . . . , n, (2.1)and say that X and Y are diagonally equivalent if (2.1) holds. Remark 2.1.
It is readily checked that AB „ diag BA for any upper (or lower) triangular matrices A and B . The associated weighted digraph G A : “ p V , E q of an n ˆ n tropical matrix A “ p a i,j q isdefined to have vertex set V : “ t , . . . , n u , and edge set E having a directed edge p i, j q P E from i to j (of weight a i,j ) whenever a i,j ‰ . A path γ is a sequence of edges p i , j q , . . . , p i m , j m q ,with j k “ i k ` for every k “ , . . . , m ´
1. We write γ : “ γ i,j to indicate that γ is a path from i “ i to j “ j m , and call γ i,s (resp. γ s,j ), where s “ i k and 1 ă k ă m, the prefix (resp. suffix )of γ i,j if γ i,j “ γ i,s ˝ γ s,j .The length ℓ p γ q of a path γ is the number of its edges. Formally, we consider also paths oflength 0, which we call empty paths . The weight w p γ q of a path γ is defined to be the tropicalproduct of the weights of all the edges p i k , j k q composing γ , counting repeated edges. The weightof an empty path is formally set to be 0.A path is simple if each vertex appears at most once. (Accordingly, an empty path is consideredalso as simple.) A path that starts and ends at the same vertex is called a cycle ; an edge ρ “ p i, i q is called a self-loop , or loop for short. We write p ρ q k for the composition ρ ˝ ¨ ¨ ¨ ˝ ρ of a loop ρ repeated k times, and call it a multiloop . The notation p ρ q is formal, and stands for an emptyloop, which can be realized as a vertex. Remark 2.2.
When a matrix A is triangular, its associated digraph G A is an acyclic digraph,possibly with loops. Since this paper concerns only with triangular matrices, in what follows weassume all graphs are acyclic digraphs .Given a path γ i,j from i to j in an acyclic digraph G A , it contains a unique simple path from i to j , denoted r γ i,j , where the remaining edges are all loops. Relabeling the vertices of G A , we mayalways assume that i ă j and thus have ℓ p r γ i,j q ď j ´ i . It is well known that powers of a tropical matrix A “ p a i,j q correspond to paths of maximalweight in the associated digraph, i.e., the p i, j q -entry of A m corresponds to the highest weight ofall the paths γ i,j from i to j of length m in G A .When dealing with product A ¨ ¨ ¨ A m of different n ˆ n matrices the situation becomes morecomplicated. Namely, we have to equip the weighted edges e i P E i of each digraph G A i with aunique color, say c i , and define the digraph G A ¨¨¨ A m : “ ď G A i , whose vertex set is t , . . . , n u and its edge set is the union of edge sets E i of G A i “ p V , E i q coloredby the c i ’s, where i “ , . . . , m . (Thus, G A ¨¨¨ A m could have multiple edges, but with differentcolors.) We called such a weighted digraph a colored digraph .Then, having such coloring, the p i, j q -entry of the matrix product B “ A ¨ ¨ ¨ A m correspondsto the highest weight of all colored paths p i , j q , . . . , p i m , j m q of length m from i “ i to j “ j m in the digraph G A ¨¨¨ A m , where each edge p i k , j k q has color c k , k “ , . . . , m , i.e., every edge iscontributed uniquely by the associated digraph G A k of A k , respecting the color ordering. We callsuch a path a proper colored path .In what follows when considering paths in colored digraphs G A ¨¨¨ A m , we always restrict to thosecolored paths that respect the sequence of coloring c , . . . , c m , i.e., properly colored, determinedby the product concatenation A ¨ ¨ ¨ A m . (For this reason we often preserve the awkward notation G A ¨¨¨ A m which records the product concatenation A ¨ ¨ ¨ A m .) ZUR IZHAKIAN
Notation 2.3.
Given a matrix product B “ A ¨ ¨ ¨ A m , we write x B y to indicate that B is realizedas a word “restoring” the product concatenation A ¨ ¨ ¨ A m , and thus denote G A ¨¨¨ A m as G x B y ,while B denotes the result of the matrix product. We also write x B y “ x B yx B y for the productconcatenation of x B y “ A ¨ ¨ ¨ A k and x B y “ A k ` ¨ ¨ ¨ A m , with ă k ă m . Semigroup identities
Semigroup elements.
Assuming that S : “ p S, ¨ q is a multiplicative monoid with identityelement e S , we write s i for the s ¨ s ¨ ¨ ¨ s with s repeated i times and formally identify s with e S .Let X be a countably infinite set of “variables” (or letters) x , x , x , . . . , i.e., X : “ t x i : i P N u .An element w of the free semigroup X ` generated by X is called a word (over X ), written uniquelyas w “ x t i ¨ ¨ ¨ x t m i m P X ` , i k P N , t k P N , (3.1)where x i k ‰ x i k ` for every k . We write κ x i p w q for the number of occurrences of the variable x i P X in the word w . Then cont p w q : “ t x i P X | κ x i p w q ě u is called the content of w and ℓ p w q : “ ÿ x i P cont p w q κ x i p w q is the length of w . A word w P X ` is said to be finite if ℓ p w q is finite. We assume that the emptyword, denoted e , belongs to X ` and set ℓ p e q “
0. A word w is called k -uniform if each letter x i P cont p w q appears in w exactly k times, i.e., κ x i p w q “ k for all x i P cont p w q . We say that w is uniform if it is k -uniform for some k .We say that w P X ` is a factor of a word w P X ` , written w | w , if w “ w w w for some w , w P X ` . When w “ w w , we call the factors w and w respectively the prefix and suffix of w , denoted as pre p w q and suf p w q . Given a word w P X ` we write pre x i p w q (resp. suf x i p w q ) forthe prefix (resp. suffix) of w of maximal length that consists only the variable x i , in particular,when w ‰ e , pre x i p w q “ x j i i for some x i and j i P Z ` .A word u is a subword of v , written u Ď v , if v can be written as v “ w u w u w ¨ ¨ ¨ u m w m where u i and w i are words (possibly empty) such that u “ u u ¨ ¨ ¨ u m , i.e., the u i are factors of u . Clearly, any factor of v is also a subword, but not conversely.Given a finite subset P Ă N , we define the “down closure” of P to be P : “ t t P N | t ď p for some p P P u . The exponent set exp p w q of a word w of the Form (3.1) is defined asexp p w q : “ t t k | t k ą u . In this paper we always assume all words are finite; thus | exp p w q| is finite for any word w . Whenexp p w q “ exp p w q “ t , . . . , m u we say that w is a word of exponent x m y . Henceforth, we always assume that P “ P Ă N is a nonempty subset of the form P : “ t , . . . , m u and that n ě m “ max t p | p P P u . Given finite nonempty subsets C Ď X and P Ă N , for any n ě m , n P N , we define W n r C, P s : “ t w P X ` | cont p w q Ď C, exp p w q Ď P, ℓ p w q “ n u , in particular W n r C, P s Ă C ` . We denote by r w p C,P,n q a word in C ` for which every member of W n r C, P s is a factor, i.e., r w p C,P,n q P C ` such that u | r w p C,P,n q for every u P W n r C, P s . (3.2) EMIGROUP IDENTITIES IN THE MONOID OF TRIANGULAR TROPICAL MATRICES 5
We call r w p C,P,n q an n - power word of C and P . We say that r w p C,P,n q is faithful ifcont p r w p C,P,n q q “ C and exp p r w p C,P,n q q “ P .
Note that ℓ p r w p C,P,n q q ě n , while for | C | ą ℓ p r w p C,P,n q q ą n . When | C | “
1, say C “ t x i u ,then x ti is an n -power word for any t ě n , and x ti is faithful only if t “ n “ max t p | p P P u . Example 3.1.
Suppose C “ t x, y u , P “ t , u . (i) When n “ we have the set W r C, P s “ t x , xy, yx, y u Ă C ` , for which r w p C,P, q “ x y x is a faithful -power (nonuniform) word of C and P of length . (ii) If n “ we get the set W r C, P s “ t x y, xyx, xy , yx , yxy, y x u Ă C ` , for which r w p C,P, q “ l is a faithful -power uniform word of C and P of length . Remark 3.2.
Given an n -power word r w p C,P,n q of C and P , it is easy to see that for any w , w P C ` , the word of the form r w C,P,n q “ w r w p C,P,n q w is also an n -power word of C and P . Therefore,taking appropriate w and w , r w p C,P,n q can be extended to a uniform n -power word. Similarextension can be performed for faithful n -power words, preserving their faithfulness.In what follows, we always work with n -power words r w p C,P,n q which are faithful and with | C | ą . An n -power word r w p C,P,n q of C and P is called a minimal n -power word if ℓ p r w p C,P,n q q ď ℓ p r w C,P,n q q for any n -power word r w C,P,n q P C ` . Example 3.3.
The power words in Example 3.1 are minimal power words.
Remark 3.4.
Given a word w P X ` , we may consider X to be a set of generic matrices A , A , . . . .Using Notation 2.3, the word w “ A i ¨ ¨ ¨ A i m can be realized as a product concatenation of matricesand thus can be written equivalently as x B y “ A i ¨ ¨ ¨ A i m , where the result of the matrix productis B “ p b i,j q . Then every entry b i,j of B corresponds to a proper colored path γ i,j from i to j oflength m in the digraph G x B y , cf. §
2. Thus, the proper coloring of all paths of length m is uniquelydetermined by w . Conversely, given a properly colored path γ i,j of length ℓ p w q in G x B y , one canrecover the word w from the coloring of the edges consisting γ i,j . Semigroup identities.
A (nontrivial) semigroup identity is a formal equality of the formΠ : u “ v, (3.3)where u and v are two different (finite) words of the Form (3.1) in the free semigroup X ` . Fora monoid identity, we allow u and v to be the empty word as well. We discuss, for simplicity,semigroup identities, but minor changes apply to monoid identities as well.A semigroup S : “ p S, ¨ q satisfies the semigroup identity (3.3) if for every morphism φ : X ` Ñ S one has φ p u q “ φ p v q . Remark 3.5.
Semigroup identities can be thought of as special case of polynomial identities (PI’s),namely as monomial identities.
ZUR IZHAKIAN
We say that an identity Π : u “ v is an n -variable identity if | cont p u q Y cont p v q| “ n . The exponent set , denoted exp p Π q , of Π is defined to be exp p u q Y exp p v q . An identity Π is said to be balanced if κ x i p u q “ κ x i p v q for every x i P X , and it is called uniformly balanced if furthermore u and v are k -uniform for some k . We define the length ℓ p Π q of Π to be ℓ p Π q : “ max t ℓ p u q , ℓ p v qu . Itis readily checked that if Π is balanced, then ℓ p u q “ ℓ p v q . Example 3.6.
Let us give some very elementary examples of semigroup identities. (i)
A commutative semigroup admits the -variable uniformly balanced identity Π : xy “ yx, whose exponent set is exp p Π q “ t u . (ii) An idempotent semigroup admits the (non-balanced) -variable identity Π : x “ x, whosecontent is cont p Π q “ t x u and its exponent set is exp p Π q “ t , u . (iii) A virtually abelian semigroup admits the -variable uniformly balanced identity Π : x n y n “ y n x n , whose exponent set is exp p Π q “ t n u . (iv) The -variable identity Π : x i y j “ y j x i , whose exponent set is exp p Π q “ t i, j u , is balancedbut not uniformly balanced for nonzero i ‰ j . An identity Π : u “ v is called a minimal identity of the semigroup S if ℓ p Π q ď ℓ p Π q for anynontrivial identity Π : u “ v of S . Remark 3.7.
Let I be a set of semigroup identities. The set of all semigroups satisfying everyidentity in I is denoted by V r I s and is called the variety of semigroups defined by I . It is easyto see that V r I s is closed under subsemigroups, homomorphic images, and direct products of itsmembers. The famous Theorem of Birkhoff says that conversely, any class of semigroups closedunder these three operations is of the form V r I s for some set of identities I . Construction of semigroup identities.
Although the main part of this paper utilizes 2-variable identities of exponent x y , for future study, we present the construction of identities thatare of our interest in full generality.Given an n -power word r w p C,P,n q as in (3.2), with C “ t x , . . . , x m u and P “ t t , . . . , t j u , we aimto build a nontrivial balanced identity Π : u “ v of content C and exponent set P . To preservethe exponent set P for Π, if necessary, we first extend r w p C,P,n q to r w C,P,n q (cf. Remark 3.2), andconstruct the words u and v such that r w C,P,n q is their prefix and suffix.Let t max : “ max t t , . . . , t j u , t min : “ min t t , . . . , t j u , and let d : “ t max ´ t min . We assume thefollowing: | C | ą , | P | ą , t max ě t min . Letting z : “ x t min ¨ ¨ ¨ x t min m , z : “ x t min m ¨ ¨ ¨ x t min , we define the identityΠ p C,P,n q : r w C,P,n q z r w C,P,n q “ r w C,P,n q z r w C,P,n q , (3.4)where r w C,P,n q is defined as r w C,P,n q : “ w r w p C,P,n q w , (3.5)with w and w given as follows (letting r w : “ r w p C,P,n q , for short): w : “ $’&’% x m if pre x p r w q ą dx if pre x m p r w q ą de otherwise w : “ $’&’% x m if suf x p r w q ą dx if suf x m p r w q ą de otherwise (3.6)Clearly, by this construction, Π p C,P,n q is a balanced identity. EMIGROUP IDENTITIES IN THE MONOID OF TRIANGULAR TROPICAL MATRICES 7
In view of Remark 3.2, r w C,P,n q can be extended further to be uniform, which then makes theidentity (3.4) uniformly balanced. In particular when the given n -power word r w p C,P,n q is uniform,we can instead explicitly define w and w in (3.5) as w : “ $&% z if pre x p r w q ą dz if pre x m p r w q ą de otherwise w : “ $&% z if suf x p r w q ą dz if suf x m p r w q ą de otherwise (3.7)to obtain a uniformly balanced identity. Notation 3.8.
Throughout this paper we use the notation x and y to mark specific instances ofthe variables x and y in a given expression, although these notations stand for the same variables x and y , respectively. Example 3.9.
Let C “ t x, y u and P “ t , u , and set z “ xy , z “ yx . (i) Starting with the -power word r w p C,P, q “ x y x of C and P given in Example 3.1.(i), bythe rule of (3.6) we extend it to r w C,P, q “ yx y x , a uniform word, and define the identity Π p C,P, q : yx y x xy yx y x “ yx y x yx yx y x . (3.8) This identity is uniformly balanced. (ii)
Taking the uniform -power word r w p C,P, q “ l of C and P as in Example 3.1.(ii), we getthe uniformly balanced identity Π p C,P, q : l xy l “ l yx l . (3.9) (In this case, by (3.6) there is no need for extension.)For both identities, Π p C,P, q and Π p C,P, q , we have cont p Π p C,P,n q q “ cont p r w C,P,n q q “ cont p r w p C,P,n q q and exp p Π p C,P,n q q “ exp p r w C,P,n q q “ exp p r w p C,P,n q q , for n “ , . Theorem 3.10.
A semigroup S : “ p S, ¨ q that satisfies an n -variable identity Π : u “ v , for n ě ,also satisfies a refined -variable identity p Π : p u “ p v of exponent x y . Proof.
Since S satisfies the n -variable identity Π : u “ v , then by definition φ p u q “ φ p v q for everymorphism φ : X ` Ñ S . Suppose C : “ cont p Π q “ t x , . . . , x n u , and write C as the disjoint union C “ C Y C , for nonempty subsets C and C . Pick two variables, say y , y , and consider thewords p u and p v , obtained respectively from u and v by substituting y y for every x i P C and y y for every x j P C . It is easy to verify that p Π : p u “ p v is a 2-variable identity, with set of exponents exp p p Π q Ď t , u . We claim that S satisfies the identity p Π : p u “ p v . Indeed, assume φ : X ` Ñ S sends φ : y ÞÑ s and φ : y ÞÑ s , then φ : y y ÞÑ s s “ a and φ : y y ÞÑ s s “ b . But, since a and b satisfy Πby hypothesis, and p u and p v can be decomposed as concatenation of the terms y y and y y , then s and s satisfy p Π. (cid:3) In the sequel, in view of Theorem 3.10, we focus on 2-variable identities Π p C,P,n q of exponent x y of the Form (3.4), with C : “ t x, y u and P : “ t , u . For ease of exposition, for a given n -powerword r w p C,P,n q we begin with the 2-variables identity having exponent set x y of the formΠ p C,P,n q : r w C,P,n q x r w C,P,n q “ r w C,P,n q y r w C,P,n q , (3.10)where here, using Remark 3.2, r w : “ r w C,P,n q is an extended n -power word, obtained by the ruleof Formula (3.6); in particular, pre x p r w q , pre y p r w q , suf x p r w q , suf y p r w q are all ď
1, which preservethe exponent x y property of Π p C,P,n q . (Note that in comparison to (3.4), the intermediate terms z i are now consisting of only one letter.) ZUR IZHAKIAN
Clearly the identity (3.10) is not balanced, however it can be easily refined by substituting x : “ q x q y and y : “ q y q x (3.11)to receive back balanced identity (of exponent x y ) q Π p q C,P,n q : q w q C,P,n q q x q y q w q C,P,n q “ q w q C,P,n q q y q x q w q C,P,n q , (3.12)with q C “ t q x, q y u , P “ t , u , and q w q C,P,n q is the word obtained from r w C,P,n q substitution (3.10).(Note that now r w C,P,n q need not be an n -power word of q C “ t q x, q y u and P “ t , u .)4. Identities of triangular tropical matrices
Aiming to prove the existence of a semigroup identity for the monoid U n p T q of n ˆ n triangularmatrices, we start with the case of diagonally equivalent matrices, which is easier to deal with;then we generalize the results to the whole monoid U n p T q . Remark 4.1.
Suppose S “ M n p T q is the monoid of all n ˆ n tropical matrices, then any semigroupidentity Π : u “ v that S admits is balanced. Indeed, otherwise assume κ x i p u q ‰ κ x i p v q for some x i and take morphism φ : x j ÞÑ I for each j ‰ i (recall that I is the identity matrix) and φ : x i ÞÑ αI for some fixed α ‰ to reach a contradiction.However, for an easy exploration, for the certain class of diagonally equivalent matrices, we firstwork with unbalanced identities for the Form (3.10) and then refine them to balanced identities asin (3.11) . When dealing with matrix identities, we sometimes denote generic matrices (standing for vari-ables x, y, . . . ) by capital letters
X, Y, . . . , as well as the words they generate. To demonstratethe main idea of our approach for proving existence of semigroups identities for tropical matrices,we first prove the existence of a semigroups identity admitted by the monoid of 2 ˆ The monoid of ˆ triangular tropical matrices. An explicit semigroup identity of thecase of 2 ˆ Theorem 4.2.
Any two matrices
X, Y P U p T q such that X „ diag Y satisfy the identity U : “ XY X XY “ XY Y XY “ : V (4.1) of the Form (3.10) .Proof. Write U “ p u i,j q and V “ p v i,j q . The equality u i,i “ v i,i is obvious for the diagonal entries.Consider the p , q -entry, say of the matrix product U , this entry corresponds to a colored path γ , from 1 to 2 of maximal weight in the digraph G XY XXY p“ G x U y q . Clearly, γ , is of length 5and it contains a simple path r γ , Ă γ , of length 1; namely the simple path r γ , is an edge. Let e X be the edge in γ , contributed by G X (or equivalently by X ). If e X is a loop, we are donesince e X can be replaced by e Y in G XY Y XY p“ G x V y q , since X „ diag Y , yielding a path of thesame weight. Otherwise, by the same argument, it is enough to show that G XY XXY has anotherpath of the same length and weight in which the contribution of G X is a loop.Assume e X is not a loop, i.e., r γ , “ e X , then γ , has the form γ , “ p ρ q ˝ e X ˝ p ρ q , EMIGROUP IDENTITIES IN THE MONOID OF TRIANGULAR TROPICAL MATRICES 9 where ρ and ρ are loops. If w p ρ q ą w p ρ q (resp. w p ρ q ą w p ρ q ) then the path p ρ q ˝ e X ˝ ρ (resp. e X ˝ p ρ q ) would have a higher weight than γ , has – a contradiction. (Note that thata loop can be contributed equivalently either by G X or G Y .) Thus w p ρ q “ w p ρ q , and hence e X ˝ p ρ q and p ρ q ˝ e X ˝ ρ are paths of the same weight as γ , in which G X contributes aloop. (cid:3) Remark 4.3.
Although we have proven Theorem 4.2 for the explicit identity (4.1) of diagonallyequivalent matrices, the same proof also holds for any identity of ˆ triangular tropical matricesgiven in the general form as in (3.10) . Corollary 4.4 ([7, Theorem 3.6]) . Any matrices
A, B P U p T q satisfy the semigroup identity AB A AB AB A “ AB A BA AB A. (4.2) Proof.
Apply Theorem 4.2 for X : “ AB and Y : “ BA as in (3.11). (cid:3) One easily sees that (4.2) is a 2-variable uniformly balanced identity of length 10. Moreover, by[7], we know that this is an identity of minimal length that U p T q satisfies.4.2. The general case.
We now turn to prove the existence of a semigroup identity in the generalcase of n ˆ n triangular tropical matrices, first generalizing Theorem 4.2 to identities of the Form(3.10) for n ˆ n diagonally equivalent triangular matrices, which later provides the proof for all n ˆ n triangular matrices. Remark 4.5.
Viewing the entries of a product of n ˆ n triangular tropical matrices as coloredpaths in the associated digraph, cf. §
2, one gets only paths containing simple subpaths of length ă n . Therefore, concerning identities of the Form (3.10) , applied to n ˆ n triangular matrices, itis enough to implement an identity Π p C,P,n ´ q constructed by using p n ´ q -power words. Given two n ˆ n triangular matrices X „ diag Y , let Z P U n p T q be the product concatenation x Z y : “ x L y X x R y , x L y “ x R y “ r w p C,P,n ´ q , (4.3)with C : “ t X, Y u and P : “ t , u . Recall that the notation X is used to mark the specific instanceof the matrix X in the expression, although it is just the same matrix as X , and that x Z y standsfor the product concatenation (i.e., a formal word) whose product result is the matrix Z “ p z i,j q given in (4.3).In the view of §
2, the p i, j q -entry z i,j of Z “ p z i,j q corresponds to a colored path γ i,j of maximalweight and length ℓ p γ i,j q “ ℓ px Z yq from i to j in the associated digraph G x Z y of x Z y . Then, for j ą i , cf. Remark 2.2, the simple colored subpath r γ i,j Ă γ i,j from i to j is of length ď j ´ i ď n ´ γ i,j contains exactly ℓ px Z yq ´ ℓ p r γ i,j q loops. Remark 4.6.
The matrix product concatenations x L y and x R y in (4.3) have been taken to be r w p C,P,n ´ q – the p n ´ q -power word of C and P for which every member of W n ´ r C, P s is afactor. This allows us to deal also with cases in which the involved matrices have diagonal entries which means that some vertices in the associated digraphs are not adjunct to a loop.In the sequel exposition, when working with matrices whose diagonal entries are all nonzero, wemay replace r w p C,P,n ´ q by a word for which every member of W n ´ r C, P s is a subword (and notnecessarily a factor), to possibly obtain shorter semigroup identities. Given a simple path r γ i,j from i to j , we write W p r γ i,j q for the word recorded uniquely by thecoloring of r γ i,j . In particular ℓ p W p r γ i,j qq “ ℓ p r γ i,j q .The next lemma plays a central role in this paper. Lemma 4.7.
Suppose x Z y is as in (4.3) , where X, Y P U n p T q . Let γ i,j , where i ă j , be a coloredpath of maximal weight in G x Z y for which the contribution of G X is a non-loop edge e X . Then G x Z y has another colored path of the same length and weight in which the contribution of G X is aloop. The proof of the lemma is quite technical, thus, before proving it formally, let us outline itsmajor idea. Given a path γ i,j from i to j , it contains a unique simple subpath r γ i,j from i to j whichcorresponds to a subword W of x Z y ; all the other edges of γ i,j are loops. We want to show that ifthe edge e X contributed by G X appears in r γ i,j , then e X can be excluded from r γ i,j by “shifting” r γ i,j in one of three ways without changing its weight: either shifting r γ i,j to the left or to the right suchthat W becomes a factor of x L y or x R y respectively, or by writing r γ i,j as a composition r γ i,k ˝ r γ k,j ,and W “ W W correspondingly, and shifting r γ i,k to the left and r γ k,j to right such that both W and W are factors of x L y and x R y respectively. As a consequence of these “shifts”, which arepossible since on each vertex G X and G Y have loops of the same weight, the contribution of G X to γ i,j becomes a loop, while the edge e X is replaced in r γ i,j by e X , contributed by another G X . Proof of Lemma 4.7.
Let m “ ℓ px R yq “ ℓ px L yq , since x L y “ x R y “ r w p C,P,n ´ q – an p n ´ q -powerword, then m ě n , and thus ℓ p γ i,j q “ m `
1. Let r m “ ℓ p r γ i,j q . In particular, r m ă n since r γ i,j is asimple path in G x Z y – an acyclic digraph on n vertices, and thus r m ă m . Recall that by hypothesis e X P r γ i,j . The proof is delivered by cases, determined by the structure of the path γ i,j .Write γ i,j “ r γ i,s ˝ p ρ s q p s ˝ p γ s,t ˝ p ρ t q p t ˝ r γ t,j , i ď s ă t ď j, (4.4)where r γ i,s (resp. r γ t,j ) is the maximal simple path (could be empty) appearing as the prefix (resp.suffix) of γ i,j , ρ s and ρ t are loops which must exist due to length considerations, and p γ s,t is asubpath (needs not be simple).Therefore, e X does not belong to r γ i,s nor r γ t,j , by length considerations, and thus ℓ p r γ i,s q , ℓ p r γ t,j q ă r m ´ p s , p t ą . Define x F y : “ W p r γ i,s q and x G y : “ W p r γ t,j q to be the words (whose terms are generic matrices)determined by the coloring of the simple paths r γ i,s and r γ t,j , and set x L y : “ x L yz pre x F y and x R y : “ x R yz suf x G y . In other words x L y and x R y are the words obtained from x L y and x R y after removing respectively the initial and the terminal words (which in this case are factors)corresponding to the simple paths r γ i,s and r γ t,j .Let µ s,t “ p ρ s q p s ˝ p γ s,t ˝ p ρ t q p t , s ă t, (4.5)be the subpath of γ i,j , given by its intermediate non-simple part according to (4.4), and let r µ s,t bethe simple subpath contained in µ s,t . Define x H y : “ W p r µ s,t q – the matrix product concatenation,realized as a subword, corresponding to the coloring of the path r µ s,t .We claim that G x L y contains a path similar to r µ s,t and G x R y contains a path similar r µ s,t . Itis enough to show that x H y is a subword of x L y . Indeed, x F yx H y Ď x L y by word construction,where x F y is the prefix of x L y by hypotheses, thus x H y Ď x L y is a subword of x L y . The case of r µ s,t Ă G x R y is dual.Let ρ max denote the loop of maximal weight in µ s,t , then we have the following possible cases:I. ρ s “ ρ max : Then, there is a path µ s,t “ p ρ max q q s ˝ r µ s,t ˝ p ρ t q q t with r µ s,t Ă G x R y and q s ą ℓ px L yq , such that w p µ s,t q “ w p µ s,t q , since otherwise we would get a contradiction tothe maximality of weight of µ s,t . Thus e X “ ρ s “ ρ max – a loop.II. ρ t “ ρ max : Then, there is a path µ s,t “ p ρ s q q s ˝ r µ s,t ˝ p ρ max q q t , with r γ s,t Ă G x L y and q t ą ℓ px R yq , such that w p µ s,t q “ w p µ s,t q , since otherwise we would get a contradiction tothe maximality of weight of µ s,t . Thus e X “ ρ t “ ρ max – a loop.III. ρ max P p γ s,t : Then, there is a path µ a,t “ p ρ s q q s ˝ r µ s,k ˝ p ρ max q q k ˝ r µ k,t ˝ p ρ q t q q t , s ă k ă t, EMIGROUP IDENTITIES IN THE MONOID OF TRIANGULAR TROPICAL MATRICES 11 with r µ s,k Ă G x L y and r µ k,t Ă G x R y such that r µ s,t “ r µ s,k ˝ r µ k,t . Thus w p µ s,t q “ w p µ s,t q ,since otherwise we would get a contradiction to the maximality of weight of µ s,t . Hence ρ k “ ρ max and e X “ ρ k – a loop.Therefore, in all the above cases we get that e X is a loop in µ s,t – a path in G x L y X x R y . Then,concatenate the simple paths r γ i,s and r γ t,j γ i,j “ r γ i,s ˝ µ s,t ˝ r γ t,j , i ď s ă t ď j, to obtain another path in G x L y X x R y for which the contribution of G X is a loop, as desired. (cid:3) Theorem 4.8.
Any two diagonally equivalent matrices
X, Y P U n p T q , i.e., X „ diag Y , satisfy theidentities Π p C,P,n ´ q of the Form (3.10) , with x “ X and y “ Y .Proof. Write X “ p x i,j q and Y “ p y i,j q . Let U “ p u i,j q and V “ p v i,j q be the matrix productsdetermined respectively by the left and the right words of the identity (3.10). We need to showthat u i,j “ v i,j for every i ď j . (The case of j ą i is trivial since x i,j “ y i,j “ ´8 , for any j ą i ,and hence u i,j “ v i,j “ ´8 .)It is easy to see that u i,i “ v i,i for every i “ , . . . , n . Assume now that i ă j , and considerthe associated colored digraphs G x U y and G x V y with matrix products U and V , realized as words x U y and x V y as given by (3.10). Assume first that u i,j ‰ ´8 , then the value of the entry u i,j corresponds to a colored path γ i,j from i to j of maximal weight and of length ℓ px U yq in the digraph G x U y . By Lemma 4.7 we may assume that the contribution of G X to γ i,j is a loop ρ X , but then G x V y also contains a similar colored path γ i,j in which the contribution of G Y is also a loop ρ Y ,replacing ρ X , which by hypothesis have the same weight, i.e., w p ρ X q “ w p ρ Y q . Dually, the sameargument also holds for a path in G x V y in which the contribution of G Y is a loop.Suppose now that u i,j “ ´8 , and assume that v i,j ‰ ´8 . This means that there exists a coloredpath γ i,j from i to j in G x V y , and in particular, by Lemma 4.7, a path in which the contributionof G Y is a loop ρ Y . But then, by the above dual argument, G x U y also has a similar path, and thus u i,j ‰ ´8 – a contradiction.Putting all together, we have u i,j “ v i,j for every i, j . (cid:3) Example 4.9.
Assume X „ diag Y , and set x “ X and y “ Y . (i) If X, Y P U p T q then they satisfy the identity Π p C,P, q : yx y x x yx y x “ yx y x y yx y x . (4.6)(ii) f X, Y P U p T q then they satisfy the identity Π p C,P, q : l x l “ l y l . (4.7) (To preserve the exponent x y of the identities, we use the power words of Example 3.1 with anadditional instance, denoted as y , of y given by the rule of (3.6) .) Theorem 4.10.
The submonoid U n p T q Ă M n p T q of upper triangular tropical matrices satisfiesthe identities q Π p C,P,n ´ q of the Form (3.12) , which we recall is (3.10) with x “ X “ AB and y “ Y “ BA , for n ˆ n generic matrices A, B .Proof.
It easy to verify that for any triangular matrices
A, B P U n p T q the matrix products X “ AB and Y “ BA are diagonally equivalent. The proof is then completed by Theorem 4.8. (cid:3) Example 4.11.
Set x “ AB and y “ BA . (i) The monoid U p T q of ˆ triangular tropical matrices satisfies the identity (4.6) . (ii) The monoid U p T q of ˆ triangular tropical matrices admits the identity (4.7) . Identity length: An upper bound
In the previous section we proved the existence of semigroup identities of the Form (3.12),satisfied by U n p T q , we now discuss the length of this identity, providing a very naive upper bound.The very well known Fibonacci sequence F n is defined by the recursive relationF n : “ F n ´ ` F n ´ , for every n ě , (5.1)where F “ “
1, and has the closed formula (known as Binet’s Fibonacci numberformula): F n “ p ` ? q n ´ p ´ ? q n n ? . (5.2)Therefore, we see that the Fibonacci number 2 F n gives the number of elements in W n r C, P s , i.e.,all possible factors of length n , for the case of C : “ t x, y u and P : “ t , u . (The multiplier 2 standsfor the two possibilities for starting a sequence, either with x or y .)A naive construction of an n -power word r w p C,P,n q , i.e., by concatenating all factors in W n r C, P s ,gives us the upper bound ℓ p r w p C,P,n q q ď p n ` q F n . (The multiplier of n `
1, stands for the length of a factor with a possible additional letter betweensequential factors, aiming to preserve the exponent x y property.) Thus, considering the refinementgiven by x “ q x q y and y “ q y q x , we have ℓ p Π p C,P,n q q ď p n ` q F n ` . (5.3)Obtusely, this rough upper bound assumes that the factors W n r C, P s do not overlap in r w p C,P,n q .Dealing with possible overlap factors, one can reduce the multiplier n ` n ˆ n triangular matrices it is enough to consideridentities Π p C,P,n ´ q , for which the upper bound is then smaller, that is ℓ p Π p C,P,n ´ q q ď n F n ´ ` . Remarks and open problems
A natural question arisen from our identity construction in § U n p T q . Conjecture 6.1.
When r w C,P,n ´ q is a minimal n -power word of P and C , then the identity q Π p C,P,n ´ q in (3.12) is a minimal semigroup identity admitted by U n p T q . The results of this paper, and those of [7], lead us to the conjecture, which has already beenconjectured earlier in [7], that
Conjecture 6.2.
Also the monoid M n p T q of n ˆ n tropical matrices satisfies a nontrivial semigroupidentity for all n . (The conjecture has been proven in [7, Theorem 3.9] for the case of n “ . )Another reason for conjecturing this is that every finite subsemigroup of M n p T q has polynomialgrowth [4, 12]. In particular, the free semigroup on 2 generators is not isomorphic to a subsemigroupof M n p T q . While Shneerson [11] has given examples of polynomial growth semigroups that do notsatisfy any nontrivial identity (no such example exists for groups by Gromov’s Theorem [5]), weconjecture that this is not the case for M n p T q . EMIGROUP IDENTITIES IN THE MONOID OF TRIANGULAR TROPICAL MATRICES 13
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Department of Mathematics, University of Bremen, 28359 Bremen, Germany.
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