Permutability of Matrices over Bipotent Semirings
aa r X i v : . [ m a t h . R A ] J a n PERMUTABILITY OF MATRICES OVERBIPOTENT SEMIRINGS
THOMAS AIRD AND MARK KAMBITES Department of Mathematics, University of Manchester,Manchester M13 9PL, UK.
Abstract.
We study permutability properties of matrix semigroupsover commutative bipotent semirings (of which the best-known exampleis the tropical semiring ). We prove that every such semigroup is weaklypermutable (a result previous stated in the literature, but with an erro-neous proof) and then proceed to study in depth the question of whenthey are strongly permutable (which turns out to depend heavily on thesemiring). Along the way we classify monogenic bipotent semirings anddescribe all isomorphisms between truncated tropical semirings.
Commutative bipotent semirings appear naturally in many areas of math-ematics; for example, the boolean semiring has important applications incomputer science [4], while tropical and related semirings have found appli-cations in areas as diverse as algebraic geometry, geometric group theory,automata and formal languages, and combinatorial optimization and controltheory [1, 2, 10]. Many of the problems which arise naturally in these areasinvolve finite systems of linear (over the semiring) equations and can there-fore be formulated in terms of matrix operations; understanding the struc-ture of matrix algebra over these semirings is thus vital for applications,and much recent research has been devoted to this topic. An additionalmotivation comes from abstract semigroup theory, where there is increasingevidence [5, 6, 7] that tropical and related semirings are natural carriersfor representations of important classes of semigroups and monoids which,due to their structural properties, do not admit faithful finite dimensionalrepresentations over fields.In this paper, we focus on two algebraic finiteness conditions for semi-groups of matrices over bipotent semirings: weak permutability and per-mutability. A semigroup S is called weakly permutable if there exists a k ≥ s , . . . , s k ∈ S there exist permutations σ = τ of { , . . . , k } such that s σ (1) s σ (2) · · · s σ ( k ) = s τ (1) s τ (2) · · · s τ ( k ) . A semigroup S is called permutable (or sometimes strongly permutable ) if there exists a k ≥ s , . . . , s k ∈ S there exists a permutation σ of { , . . . , k } such that s σ (1) s σ (2) · · · s σ ( k ) = s s · · · s k . We note here a few key facts about theseproperties; for a comprehensive introduction the reader is directed to [11,Chapter 19]. Notice that every strongly permutable semigroup S is weakly Date : January 12, 2021.
Key words and phrases. semirings, matrices, permutability, tropical matrices. Email
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[email protected] . permutable by taking τ to be the identity permutation. Every finite semi-group is clearly strongly permutable, as is every commutative semigroup.Indeed, weak and strong permutability may be thought of as very weakcommutativity conditions. It is easy to see that if a semigroup S is weakly[strongly] permutable then every subsemigroup of S and every homomorphicimage of S is also weakly [strongly] permutable. Permutability conditionsare of interest in general because of connections with polynomial identitiesin semigroup algebras [11, Chapter 19], and are lent additional importancein these particular semigroups by interest in representations over semirings:any permutability condition satisfied by matrix semigroups poses an ob-struction to faithfully representing semigroups not satisfying the condition.We begin, in Section 1, by establishing some structural results about com-mutative bipotent semirings which will be useful in our subsequent analysis.These include a simple classification of the monogenic examples, which maybe of independent interest.In Section 2 we proceed to look at weak permutability, proving that everyfull matrix semigroup, and hence every matrix semigroup, over a commuta-tive bipotent semiring is weakly permutable. This fact was first stated byd’Alessandro and Pasku [3] but there is an error (described below) in theirproof.In Section 3 we turn our attention to (strong) permutability. If the semir-ing has an element of infinite multiplicative order (or more generally, el-ements of unbounded multiplicative order) we prove (Theorem 3.6) thatthe full matrix semigroup and upper triangular matrix semigroups are notstrongly permutable in any dimension greater than 1. This applies in par-ticular to the tropical and many related semirings. On the other hand,semirings with bounded multiplicative order exhibit a range of behaviours,with apparently similar semirings sometimes differing quite dramatically.Matrix semigroups over chain semirings , which are multiplicatively as wellas additively idempotent, are strongly permutable in all dimensions (Corol-lary 3.10).Section 4 is devoted to the class of truncated tropical semirings , whereit transpires that the full matrix semigroups can be strongly permutablein all dimensions (Theorem 4.3), only in dimension 1 (Corollary 4.2) or,interestingly, only in dimensions 1 and 2 (Theorem 4.7). Similar resultsare obtained for the monoids of upper triangular and upper unitriangularmatrices. In the course of our study we describe all isomorphisms betweentruncated tropical semirings.Throughout this paper we write N for the set of natural numbers excluding0. For n ∈ N we write [ n ] for the discrete interval N ∩ [1 , n ], and S n for thesymmetric group on the set [ n ]. Acknowledgements.
The authors thank Marianne Johnson for somehelpful conversations and comments on the draft.1.
Commutative Bipotent Semirings
For our purposes a semiring S is a non-empty set with two binary op-erations — addition and multiplication — such that both operations areassociative, addition is commutative, and multiplication distributes over ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 3 addition on both sides. A semiring is called commutative if the multipli-cation is commutative and bipotent if x + y is always either x or y . Abipotent semiring admits a natural linear order defined by x ≤ y if and onlyif x + y = y , and the distributive laws mean exactly that multiplicationrespects this order, giving rise to a totally ordered semigroup. Conversely,every totally ordered semigroup gives rise to a bipotent semiring, by takingthe semigroup operation as multiplication and defining the sum to be max-imum with respect to the order. Bipotent semirings are thus, at one level,the same thing as totally ordered semigroups, but the two viewpoints leadnaturally to rather different questions; in particular the semiring viewpointleads to the study of linear algebra and matrices. Our main interest is incommutative bipotent semirings, although some of our results will extendto the non-commutative case.Some authors insist that a semiring should have a zero (that is an elementwhich is a multiplicative zero and an additive identity) and/or a (multiplica-tive) identity element, but most of our results will not require these. In factit is easy to see that any commutative bipotent semiring S without a zero el-ement can have one “adjoined”, that is, can be embedded in a commutativebipotent semiring with one extra element 0 which is a zero. We write S forthis semiring; if S already has a zero element then we define S = S and use0 to denote the zero element of S . On the other hand, the correspondingstatement is not true for identity elements: Proposition 1.1.
There exists a commutative bipotent semiring S withoutidentity which cannot be embedded in any bipotent semiring with identity.Proof. Let S = { a, b, c } be the commutative bipotent semiring such that c ≥ b ≥ a , all elements are multiplicatively idempotent, and all non-idempotentproducts are b . It is straightforward to verify that the given operationsrespect the associative and distributive laws. Now suppose we can embed S in a bipotent semiring with identity 1, and consider where 1 lies in theorder. If 1 > b , then a (1 + b ) = a a , but by the distributive law a (1 + b ) = a ab = a + b = b giving a contradiction. On the other hand,if 1 < b , then c (1 + b ) = cb = b , but similarly by the distributive law c (1 + b ) = c cb = c + b = c , giving a contradiction. Thus, we cannotembed S into a bipotent semiring with identity. (cid:3) Notwithstanding the impossibility in general of adjoining an identity el-ement, it is sometimes convenient to introduce “the identity” as a purelynotational device. If S is a commutative semiring without identity, we de-fine S to be S ∪ { } where 1 is a new symbol, and define 1 x = x for all x and 1 + 1 = 1 and also (where S has a 0) 1 + 0 = 1, but leave other sumsinvolving 1 undefined. We caution that this structure is not a semiring, sinceaddition is only partially defined. Again, if S already has an identity we set S = S and use 1 to denote the existing identity. We write S for ( S ) .A subsemiring is a subset closed under addition and multiplication; notethat even if S has zero and/or identity elements, subsemirings are not re-quired to contain them. If a ∈ S then we write h a i for the ( monogenic )subsemiring of S generated by a (that is, the intersection of all subsemiringscontaining a ). If S is bipotent then h a i coincides with the multiplicative PERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS subsemigroup of S generated by a , in other words, the set of positive powersof a . The (multiplicative) order of a is defined to be the cardinality of theset of positive powers of a , which when S is bipotent is the cardinality of h a i .We will consider in particular the following examples of commutativebipotent semirings; some of these merit study due to external applications,some arise naturally in the general theory, and others are included to illus-trate the full range of possible behaviours: • The tropical (or max-plus ) semifield T consists of the real numbersaugmented with −∞ , with maximum as its addition and addition asits multiplication; it has applications in numerous areas including;biology [1], control theory [2] and algebraic geometry [10]. The tropi-cal semifield admits isomorphic manifestations as the min-plus semi-field (the real numbers augmented with + ∞ under minimum andclassical addition) and the max-times semifield (the non-negativereal numbers under maximum and classical multiplication ). • The tropical natural number semiring N max is the subsemiring of T consisting of natural numbers; it has applications in areas such asformal language theory and automata theory [9]. • The tropical negative natural number semiring ( − N ) max is the sub-semiring of T consisting of the negative integers. (It is isomorphicto the natural numbers under minimum and classical addition.) • For x, y ∈ R with 0 ≤ x < y the truncated tropical semiring T [ x,y ] consists of the real interval [ x, y ] augmented with 0 and −∞ with op-erations maximum and y -truncated addition given by ab = min( a + b, y ) where + here denotes classical addition. • For k ∈ N the truncated tropical natural number semiring [ k ] max consists of the set [ k ] = { , . . . , k } with operations defined as in T [1 ,k ] . • For k ∈ N the truncated tropical negative natural number semiring [ − k ] max consists of the set {− k, . . . , − } with operations maximumand ( − k ) -truncated addition given by a · b = max( a + b, − k ). (Notethat [ − max and [1] max are both trivial and therefore isomorphic toeach other.) • Any linearly ordered set admits the structure of a commutative bipo-tent semiring, with maximum as addition and minimum as multipli-cation. We call these chain semirings . A prominent example is the2-element chain semiring, the boolean semifield , which is isomorphicto the semiring with two elements
T rue and
F alse with operations“and” and “or”, and has natural applications in logic and computerscience [4].For any semiring S and n ∈ N , the set of n × n matrices over S formsa semiring under matrix multiplication and addition induced from S in theusual way. Note that M n ( S ) will typically be neither commutative norbipotent (even when S is both). Our main interest here is in the multiplica-tive semigroup of M n ( S ). We also define U T n ( S ) to be the subsemiring of M n ( S ) consisting of matrices with 0 below the main diagonal and elementsfrom S on and above the main diagonal. We write U n ( S ) for the semiring ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 5 of matrices over S which have 0 below the main diagonal, 1 on the maindiagonal and elements from S above the main diagonal; note that even ifthe 1 is adjoined the partial addition in S is defined for sufficiently manyvalues to enable matrix addition and multiplication on this set. Again, ourprincipal interest is in the structure of U T n ( S ) and U n ( S ) as multiplicativesemigroups. Note that M ( S ) = U T ( S ) is isomorphic to the multiplicativesemigroup of S , while U ( S ) is the trivial monoid and U ( S ) is isomorphicto the additive semigroup of S . Lemma 1.2.
Let S be a bipotent semiring. If an element x ∈ S has finitemultiplicative order (that is, has finitely many distinct powers) then it hasperiod (that is, x k = x k +1 for some k ∈ N ).Proof. Let x ∈ S have finite multiplicative order. Then there exist r, m ∈ N such that x m = x m + r . If r = 1 we are done, so assume r >
1. As S isbipotent we have that the sum x m + · · · + x m + r − = x k for some k between m and m + r −
1. But now by distributivity and commutativity of addition, x k +1 = x ( x m + · · · + x m + r − + x m + r − )= x m +1 + · · · + x m + r − + x m = x k . (cid:3) The following lemma describes all the possible bipotent semirings gener-ated by a single element:
Lemma 1.3.
Let S be a bipotent semiring. If a ∈ S and h a i is the monogenicsubsemiring generated by a , then h a i ∼ = N max if a has infinite order and a < a ;( − N ) max if a has infinite order and a < a ;[ k ] max if a has order k ∈ N and a ≤ a ;[ − k ] max if a has order k ∈ N and a ≤ a. We remark that the four cases above are comprehensive but not quitemutually exclusive: in the case that a has order 1 we have a = a and h a i isisomorphic to both [1] max and [ − max . Proof.
First suppose a ≤ a . Define a map φ : N max → h a i , n a n . This map is surjective (because of our observation that, in a bipotent semir-ing, h a i coincides with the multiplicative semigroup generated by a ) and pre-serves multiplication because of basic properties of powers. Now let n, m ∈ N and suppose without loss of generality that n ≥ m . Since a ≤ a we have a k ≤ a k +1 for all k (because the total order is compatible with multipli-cation) and hence a m ≤ a n (because m ≤ n and the order is transitive).Therefore φ (max( n, m )) = φ ( n ) = a n = a n + a m = φ ( n ) + φ ( m ) . If a has infinite order then φ is injective, and we have shown that it is anisomorphism from N max to h a i . PERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS If a has finite order k then let ϕ be the restriction of φ to the subset [ k ].Clearly ϕ is a bijection. Since the semiring addition (in other words, theorder) on [ k ] max is the restriction of that on N max , the fact that ϕ preservessemiring addition follows from the fact that φ does. Now let n, m ∈ N andsuppose without loss of generality that n ≥ m . Then ϕ ( nm ) = a n + m = a n a m = ϕ ( n ) ϕ ( m )for all n, m ∈ N . The first equality here holds because if n + m ≥ k then a n + m = a k , as a has period 1 by Lemma 1.2. Hence, ϕ is an isomorphismbetween [ k ] max and h a i .Similarly if a ≤ a then we define ψ : ( − N ) max → h a i , n a − n . Again ψ is surjective. This time for negative integers n ≥ m we use a ≤ a to deduce that a − m ≤ a − n so ψ (max( n, m )) = ψ ( n ) = a − n = a − n + a − m = ψ ( n ) + ψ ( m ) . and ψ preserves semiring addition. If a has infinite order then ψ is injectiveand preserves the semiring multiplication, so it is an isomorphism between( − N ) max and h a i . If a has finite order k then an entirely similar argumentto that above shows that the restriction of ψ to − [ k ] is an isomorphismbetween [ − k ] max and h a i . (cid:3) Weak Permutability
In this section we briefly consider weak permutability, showing that anysemigroup of matrices over a commutative bipotent semiring always has thisproperty. This result was first stated by d’Allesandro and Pasku [3], butTaylor [12] identified an error in their proof. The error and its consequencesare discussed below. Our proof is, nonetheless, inspired by their method.
Proposition 2.1.
Let S be a commutative bipotent semiring. Then M n ( S ) is weakly permutable for all n ∈ N .Proof. Fix n ∈ N . Let Γ n denote the complete directed graph (with loops)on the set [ n ]. We identify edges in Γ n with pairs in [ n ] × [ n ] in the obviousway; in particular we will index the entries of n × n matrices by edges in Γ n .Let Π denote the set of n × n matrices whose entries are edges from Γ n (that is, pairs from [ n ] × [ n ]). Let c = | Π | = n n . Choose k large enoughthat k ! > c k .Consider a finite sequence of k matrices of size n × n over the semiring S ,say M , . . . , M k . For a permutation σ in the symmetric group S k , write M σ = M σ (1) M σ (2) M σ (3) · · · M σ ( k ) . We must show that there are distinct permutations σ, τ ∈ S k with M σ = M τ .We define a function π : S k → Π [ k ] (where Π [ k ] denotes the set of functionsfrom [ k ] to Π) as follows. For each σ ∈ S k and each x, y ∈ [ n ], considerthe ( x, y ) entry of the matrix M σ . It follows from the definition of matrixmultiplication and the fact S is bipotent that there is at least one path p , . . . , p k of length k from x to y in Γ n such that this entry is given by( M σ ) x,y = ( M σ (1) ) p ( M σ (2) ) p · · · ( M σ ( k ) ) p k . (1) ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 7
Choose any such path, and for each i ∈ [ k ] define the ( x, y ) entry of ( π ( σ )) ( i )to be the edge p σ − ( i ) (that is, the edge indexing the entry of M i whichcontributes in the computation of the ( x, y ) entry of M σ ). Thus reorderingthe terms in (1) we have( M σ ) x,y = ( M ) ( π ( σ ))(1) x,y ( M ) ( π ( σ ))(2) x,y · · · ( M k ) ( π ( σ ))( k ) x,y But this means that M σ is a function of π ( σ ).The domain S k of π has cardinality k ! while the codomain Π [ k ] of π hascardinality | Π | k = c k . Since k was chosen such that k ! > c k there mustbe distinct permutations σ, τ ∈ S k such that π ( σ ) = π ( τ ), which by theprevious paragraph means that M σ = M τ . (cid:3) The mistake in [3] lies in the proof of the first part of [3, Proposition3], where k is taken to be the smallest integer such that αk β < k !. Theproblem is that k was discussed prior to this point, and in fact played animplicit role in the definition of the set C , the cardinality of which was inturn used to define α and β . Thus, one is not necessarily free to choose k atthis point without also changing α and β . The claim that one may choose k with αk β < k ! implicitly assumes α and β to be constant, when in realitythey are functions of k and there is no immediate reason to suppose that αk β grows more slowly than k !.We discuss briefly the impact upon the correctness of other results in[3]. The second part of [3, Proposition 3] (which establishes the very im-portant result that finitely generated semigroups of tropical matrices havepolynomial growth) is correct, even though the proof ostensibly employs thesame argument as the first part; the erroneous section of the argument isnot required in this part, and the values of α and β (and hence also of δ and γ ) here are independent of k so that the growth bound obtained reallyis polynomial in k . [3, Proposition 4] is claimed to be proved by “a slightgeneralisation” of the (erroneous) proof of [3, Proposition 3]; we believe avariation on the above proof technique can be used to establish this result,but we do not do this here as it is (not being concerned with bipotent semir-ings) rather outside the scope of the present paper. The statement of [3,Proposition 5] is true: the main proof given relies on [3, Proposition 4] andis therefore incomplete, but the alternative proof via Gromov’s polynomialgrowth theorem, outlined in [3, Remark 3], is valid.3. Strong Permutability
In this section we turn our attention to the stronger version of permutabil-ity. We shall need the following result, which is trivial where the semiring S has a zero element but requires slightly more work when it does not. First,recall that for a product of k matrices M · · · M k and a permutation σ ∈ S k ,we write M σ = M σ (1) · · · M σ ( k ) . Proposition 3.1.
Let S be a commutative bipotent semiring. If M n ( S ) isstrongly permutable then M m ( S ) is strongly permutable for all m < n . If U T n ( S ) is strongly permutable then U T m ( S ) is strongly permutable for all m < n . PERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS
Proof.
Consider first the case of full matrix semigroups. Suppose false fora contradiction; then there is an m < n such that for every k ∈ N thereexist m × m matrices M , . . . , M k such that M σ = M e for any non-trivialpermutation σ . Fix k and let M , . . . , M k be as given. Let z be the smallest(with respect to the order on the semiring) entry of any matrix M i . For each i let N i be the n × n matrix obtained by taking M i and adjoining n − m rows at the bottom and n − m columns at the right in which every entry is z . Now consider the x, y entry of a product N i · · · N i k for x, y ≤ m . As S is bipotent this entry is equal to the maximum (with respect to the order inthe semiring) across sequences x = x , x , . . . , x k = y of the term: k Y j =1 ( N i j ) x j − ,x j . If in such a sequence we have x j > m for some 1 ≤ j < k , then ( N i j ) x j − ,m ≥ z = ( N i j ) x j − ,x j and ( N i j +1 ) m,x j +1 ≥ z = ( N i j +1 ) x j ,x j +1 by definition, sowe may replace x j by m in the sequence without reducing the resultingterm. Thus, we may assume the above maximum is attained for a sequencewith x j ≤ m for all j , and it follows that the top-left m × m submatrix ofthe product is the product of the corresponding submatrices in the factors,in other words, the corresponding product of the M i s. In particular, forany permutation σ the top-left m × m submatrix of N σ is exactly M σ .Thus, N σ = N e for any non-trivial permutation σ , which since k was chosenarbitrarily contradicts the assumption that M n ( S ) is permutable.For the upper triangular case, there exists a surjective homomorphismfrom U T n ( S ) to U T m ( S ) for m < n by only considering the first m rows andcolumns. Hence if U T n ( S ) is permutable then U T m ( S ) is permutable for all m < n . (cid:3) Our next objective is to show that matrix semigroups over a (not necessar-ily commutative) bipotent semiring with elements of infinite multiplicativeorder (or more generally, unbounded multiplicative order) are not, in gen-eral, permutable. A key tool is a result of Okni´nski [11, Chapter 19, Lemma22], stating that a finitely generated inverse semigroup with infinitely manyidempotents cannot be permutable. In particular this means that the bi-cyclic monoid is not permutable. This will combine with a representationof the bicyclic monoid by tropical matrices, due to Izhakian and Margolis[6], to yield non-permutability results for tropical matrix monoids, and thenwith our classification of the monogenic bipotent semirings (Lemma 1.3) toobtain non-permutability results for matrix monoids over semirings with ele-ments of infinite order. Some elementary model theory extends these resultsto semirings with unbounded order.
Theorem 3.2. M n ( N max ) , M n (( − N ) max ) , U T n ( N max ) and U T n (( − N ) max ) are not strongly permutable for n ≥ .Proof. Let B = h p, q | pq = 1 i be the bicyclic monoid. Recall that everyelement of B can be written as q i p j for some i, j ∈ N ∪ { } . By [6] there is ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 9 a semigroup embedding of B into U T ( T ) given by ρ : B →
U T ( T ) , q i p j (cid:18) i − j i + j −∞ j − i (cid:19) . Since the bicyclic monoid is not permutable [11, Chapter 19, Lemma 22]and subsemigroups of permutable semigroups are permutable, we deducethat
U T ( T ) is not permutable. Indeed further, for every k there are uppertriangular matrices M , . . . , M k whose diagonal and above-diagonal entriesare integers, with the property that M σ = M e for every non-trivial permu-tation σ ∈ S k .If we fix an integer λ strictly less then every integer appearing in thesematrices, then the tropically scaled matrices ( − λ ) M , . . . , ( − λ ) M k clearlyalso have this property. Replacing the −∞ entry of these matrices with thezero element of N yields a sequences of matrices to show that U T ( N max )is not strongly permutable. Similarly, tropically scaling M , . . . , M k by thenegative of an integer greater than every entry yields a sequence of matricesfor each k showing that U T (( − N ) max ) is not strongly permutable.It remains to establish the claims for full matrix semigroups. (Note that,since the semirings here lack zero elements, we do not have a natural em-bedding of each upper triangular matrix semigroup into the correspondingfull matrix semigroup which would allow us to immediately deduce the re-maining claims.)Let k > M , . . . , M k be as above. Choose a very large µ ∈ N ,and let N , . . . , N k ∈ M n ( N max ) be obtained from M , . . . , M k by scalingtropically by µ , and replacing the −∞ below the diagonal with 1. Nowconsider the product N σ for some σ ∈ S k , and in particular the computationof the ( x, y ) entry for some ( x, y ) = (2 , µ was chosen large enough, the terms which do not feature the(2,1) entry of any N i will all exceed those which do, from which it followsthat ( N σ ) x,y = kµ + ( M σ ) x,y . Thus, we conclude that N σ = N e . Since k and σ were arbitrary, this means that M n ( N max ) is not strongly permutable.Finally, tropically scaling the matrices N , . . . , N k by a sufficiently nega-tive integer gives a sequence to show that M n (( − N ) max ) is not strong per-mutable (cid:3) Lemma 3.3. U n ( N max ) is strongly permutable if and only if n ≤ .Proof. Recall that U ( N max ) is trivial while U ( N max ) is isomorphic to the(commutative) additive semigroup of the semiring, so both are strongly per-mutable. There exists a surjective morphism from U n ( N max ) to U ( N max ) forall n ≥ U ( N max ) is not strongly permutable.So, we define the sequence of matrices B , B , . . . , B m by B i = i m −∞ m + 1 − i −∞ −∞ (Note that technically speaking −∞ , / ∈ N max ; the “ −∞ ” and “0” featuredhere are technically the zero and identity elements adjoined in ( N max ) which is used in the definition of the unitriangular matrix semigroup U ( N max ), but because this is essentially the same as the subsemiring N max ∪ { , −∞} of T it is clearer to denote them in this way.) A simple inductive argumentshows that for each k , k Y i =1 B i = k m −∞ m −∞ −∞ Now, suppose σ ∈ S m is such that B σ := Q mi =1 B σ ( i ) = Q mi =1 B i . By thedefinition of matrix multiplication, for any j < k we must have m = ( B σ ) , ≥ ( B σ ( j ) ) , + ( B σ ( k ) ) , = σ ( j ) + m + 1 − σ ( k )and hence σ ( j ) < σ ( k ). Since σ is a permutation, this can only happen if σ is the identity permutation. Further, as m was arbitrary no non-trivialpermutations preserve this product for any m ∈ N , so U ( N max ) is notstrongly permutable. (cid:3) Lemma 3.4. U n (( − N ) max ) is strongly permutable if and only if n ≤ .Proof. Much as in the previous proof, U (( − N ) max ) is the trivial monoidwhile U (( − N ) max ) is isomorphic to the (commutative) additive semigroupof the semiring, so both are clearly strongly permutable, and there is asurjective morphism from U n (( − N ) max ) to U (( − N ) max ) for all n ≥
3, so itsuffices to show that U (( − N ) max ) is not strongly permutable.To this end we define the sequence of matrices C , . . . , C m given by C i = i − m − − m − −∞ − i −∞ −∞ Once again, the −∞ and 0 here are formally speaking the zero and iden-tity elements in (( − N ) max ) . The product of the first k such matrices isinductively seen to be k Y i =1 C i = k − m − − m − −∞ − −∞ −∞ Now, if σ ∈ S m is such that C σ := Q mi =1 C σ ( i ) = Q mi =1 C i then for any j < k ,( C σ ) , = − m − ≥ ( C σ ( j ) ) , + ( C σ ( k ) ) , = σ ( j ) − m − − σ ( k )so that σ ( j ) < σ ( k ). Since σ is a permutation, this can only happen if σ is the identity permutation. Further, as m was arbitrary no non-trivialpermutations preserve this product for any m ∈ N , so U (( − N ) max ) is notstrongly permutable. (cid:3) Lemma 3.5.
Let S be a (not necessarily commutative) bipotent semiring.If S has an element of infinite multiplicative order, then M n ( S ) and U T n ( S ) are not strongly permutable for n ≥ and U n ( S ) is not strongly permutableif and only if n ≥ .Proof. Suppose a ∈ S has infinite order. Then by Lemma 1.3 we have thatsubsemiring generated by a is isomorphic N max or ( − N ) max . Hence, M n ( S )contains an embedded copy either of M n ( N max ) or of M n (( − N ) max ); sinceneither of these are permutable for n ≥ M n ( S ) is not ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 11 permutable for n ≥
2. Similarly,
U T n ( S ) is not permutable for n ≥ U n ( S ) is not permutable if and only if n ≥ (cid:3) A bipotent semiring (even a commutative one) may have elements ofunbounded finite order, without having an element of infinite order. Forexample, we shall see below that the truncated tropical semiring T [0 , issuch a semiring. Some basic model theory allows us to extend the aboveresult to this case; we direct the reader unfamiliar with model theoretictechniques to for example [8]. Theorem 3.6.
Let S be a (not necessarily commutative) bipotent semiringwith elements of unbounded multiplicative order (that is, such that for all k ∈ N there exists an x ∈ S such that x has multiplicative order greater than k ). Then the semigroups M n ( S ) and U T n ( S ) are not strongly permutablefor n ≥ . The semigroup U n ( S ) is not strongly permutable if and only if n ≥ .Proof. Consider the set of first-order sentences in the language of semirings: L = { x m = x n | m, n ∈ N , m = n } where x is a variable and x m is shorthand for the product of m copies of x . Since S has elements of unbounded order, L is finitely satisfiable (everyfinite subset of L holds for some x ∈ S ) which means that L is a 1-type of S . By realisability of types (see for example [8, Lemma 23.6]) there existsan elementary extension of S (a structure containing S and satisfying ex-actly the same first-order theory) in which L is satisfiable, that is, in whichthere is an element x satisfying all of the sentences in L . Let T be such astructure and x ∈ T such an element. The axioms for a bipotent semiringare clearly all expressible as first-order sentences, so the structure T is itselfa bipotent semiring. Moreover, since x satisfies all sentences in L , x is anelement of infinite order, and so by Lemma 3.5 we deduce that M n ( T ) is notpermutable.Now suppose for a contradiction that M n ( S ) was permutable. This meansthere exists an m such that ∀ X , . . . , X m ∈ M ( S ) , _ σ ∈S m \{ m } X · · · X m = X σ (1) · · · X σ ( m ) . Since matrix multiplication is first-order definable in the language of semir-ings, this can clearly be re-expressed as a first-order sentence over S , featur-ing mn universally quantified scalar variables corresponding to the entriesof the m matrices. But T is elementary equivalent to S , so this sentencealso holds in T , which contradicts the fact that M n ( T ) is not permutable.Near-identical arguments show that U T n ( S ) is not permutable for n ≥ U n ( S ) is not permutable for n ≥
3. Finally, recall that U ( S ) istrivial while U ( S ) is isomorphic to the additive semigroup of S , which isalways commutative and hence strongly permutable. (cid:3) Recall that M ( S ) = U T ( S ) is isomorphic to the multiplicative semi-group of the semiring S . This may be permutable (for example when the semiring is commutative) or non-permutable (for example when S is a non-commutative free monoid with a bipotent addition given by the shortlextotal ordering). Corollary 3.7.
Let S be a commutative bipotent semiring with unboundedorder. Then M n ( S ) (and U T n ( S ) ) are strongly permutable if and only if n = 1 . Recall that a semifield is a commutative semiring, possibly without zero,where the non-zero elements form a group with multiplication. In the caseof semifields, we can now give an explicit description of when the matrixsemigroups are permutable.
Corollary 3.8.
Let S be a bipotent semifield. Then M n ( S ) and U T n ( S ) arepermutable for n ≥ (and U n ( S ) is permutable for n ≥ ) if and only if S is the 2-element boolean semifield.Proof. Since S is a bipotent semiring we have that every element has infiniteorder or period 1 by Lemma 1.2. However S is a semifield, so the non-zeroelements form a group with multiplication so the only possible elements ofperiod 1 are the identity and the zero if there is one. Thus, non-identity,non-zero elements are of infinite order. Therefore if S is not the 2-elementboolean semifield, it must have an element of infinite order and thus byTheorem 3.6 (or Lemma 3.5), M n ( S ) and U T n ( S ) are not permutable for n ≥ U n ( S ) is not permutable for n ≥
3. If B is the 2-element booleansemifield then M n ( B ) , U T n ( B ), and U n ( B ) are finite and hence permutablefor all n ∈ N . (cid:3) Theorem 3.9.
Suppose S is a (not necessarily commutative or bipotent)semiring with the following property: for every finite subset X , there existsa homomorphism to a finite semiring of order bounded by a function in thesize of X such that each element of X occupies its own singleton kernelclass. Then M n ( S ) is permutable for all n ∈ N .Proof. Let k be such that for every subset X of S with | X | = n , there is ahomomorphism from S to a finite semiring of size at most k such that eachelement of X occupies its own kernel class. Let m = k n + 1, and supposeΣ = A A · · · A m = x , . . . x ,n ... . . . ... x n, . . . x n,n By assumption we may choose a semiring homomorphism φ mapping S intoa semiring F of cardinality at most k , such that each x i,j occupies its ownkernel class. From this semiring homomorphism, we define a semigrouphomomorphism ψ mapping M n ( S ) into M n ( F ) where( ψ ( A )) i,j = φ ( A i,j ) for all i, j. Notice that, since the entries of Σ each occupy their own φ -kernel class, Σoccupies its own ψ -kernel class. Since F has cardinality at most k , M n ( F )has cardinality at most k n < m , so there must exist distinct i and j with ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 13 ψ ( A i ) = ψ ( A j ). Let σ ∈ S m be the transposition swapping i and j . Thenclearly ψ ( A σ (1) . . . A σ ( m ) ) = ψ ( A σ (1) ) . . . ψ ( A σ ( m ) ) = ψ ( A ) . . . ψ ( A m ) = ψ (Σ) , which since Σ occupies its own ψ -kernel class means that A σ (1) . . . A σ ( m ) = Σ = A . . . A m , as required to show that M n ( S ) is permutable. (cid:3) Recall that we say a binary relation ∼ = on a semiring is a congruence if ∼ =is an equivalence relation and if a ∼ = b and c ∼ = d together imply that ac ∼ = bd and a + c ∼ = b + d . Corollary 3.10.
Let S be a chain semiring (that is, a totally ordered setwith operations maximum and minimum). Then M n ( S ) is permutable forall n ∈ N .Proof. Let X = { x , . . . , x k } be a finite subset of S . Define a binary relation ≡ on S by a ≡ b if and only if a and b either (i) are equal or (ii) are not in X and lie above exactly the same elements of X . Recalling that S is totallyordered, it is easy to see that ≡ is an equivalence relation with at most2 | X | + 1 classes (being the singleton sets containing elements of X , and theopen order intervals above, below and between elements of X ), in which eachelement of X occupies its own equivalence class. Further, it can be readilyseen that ≡ is a congruence. Hence, by the usual first isomorphism theorem,the natural morphism S → S/ ≡ satisfies the conditions of Theorem 3.9. (cid:3) Truncated Tropical Semirings
In this section we shall illustrate some of the “wilder” behaviour which ispossible in commutative bipotent semirings, by studying truncated tropicalsemirings. To avoid confusion with classical operations, which we shall alsoneed, we use the symbols ⊕ and ⊗ for the denote the addition (maximum)and multiplication (truncated addition) operations in a truncated tropicalsemiring. The symbol + and juxtaposition will be used for standard arith-metic addition and multiplication of real numbers, respectively. We beginby observing that there are a number of isomorphisms between semirings inthis class: Theorem 4.1.
Let y > x ≥ . Then T [ x,y ] ∼ = T [0 , if x = 0; T [1 , if x > and y ≤ x ; T [1 , . if x > and x < y < x ; T [1 , yx ] if x > and y ≥ x. The semirings T [0 , , T [1 , , T [1 , . and T [1 ,y ] for y ≥ are pairwise non-isomorphic.Proof. If x = 0, we define the map φ : T [0 ,y ] → T [0 , by φ ( −∞ ) = −∞ and φ ( z ) = zy for z ∈ [0 , y ] . As classical multiplication distributes over classical addition, and the factthat y > φ is order preserving, it can be easily seen that φ isan isomorphism.If x > y ≤ x , we define the map φ : T [ x,y ] → T [1 , by φ ( −∞ ) = −∞ , φ (0) = 0 , and φ ( z ) = z − xy − x + 1 for z ∈ [ x, y ] . Now, for a, b ∈ [ x, y ], we have that φ ( a ) ⊗ φ ( b ) = min (cid:18) a − xy − x + 1 + b − xy − x + 1 , (cid:19) = 2 = φ ( a ⊗ b )as a, b ≥ x . Moreover, as y − x > φ is order preserving. Hence, it can beeasily seen that φ is an isomorphism.If x > x < y < x , we define a piecewise linear map φ : T [ x,y ] → T [1 , . by φ ( z ) = z − x y − x ) + 2 if 2 x ≤ z ≤ y z − ( y − x )2(3 x − y ) + 1 . y − x < z < x z − x y − x ) + 1 if x ≤ z ≤ y − x z = 0 −∞ if z = −∞ Now, for a ∈ [ y − x, y ] and b ∈ [ x, y ], we have that φ ( a ) ⊗ φ ( b ) = 2 . φ ( y ) = φ ( a ⊗ b )as φ ( a ) ≥ . φ ( b ) ≥
1. Finally, if a, b ∈ [ x, y − x ] then φ ( a ) ⊗ φ ( b ) = min (cid:18) a − x y − x ) + 1 + b − x y − x ) + 1 , . (cid:19) = min (cid:18) ( a + b ) − x y − x ) + 2 , y − x y − x ) + 2 (cid:19) = min( a + b, y ) − x y − x ) + 2= φ ( a ⊗ b )as a ⊗ b ≥ x . Moreover, as y − x > x − y > φ isorder preserving, and hence it can be easily seen that φ is an isomorphism.If x > y > φ from T [ x,y ] to T [1 , yx ] by φ ( −∞ ) = −∞ , φ (0) = 0 , and φ ( z ) = zx for z ∈ [ x, y ] . As classical multiplication distributes over classical addition, and that x > φ is order preserving, it can be easily seen that φ is an isomor-phism.For non-isomorphism we have to show that T [0 , , T [1 , , T [1 , . and T [1 ,y ] for y ≥ T [0 , is not isomor-phic to any of the others, as it is the only one with unbounded multiplicativeorder. Similarly, T [1 ,y ] has no elements of multiplicative order 3 if and onlyif y ≤ T [1 , is not isomorphic to the others. For T [1 , . , note that T [1 ,y ] has no elements of multiplicative order 4 if and only if y ≤
3, so T [1 , . cannot be isomorphic to any of the others apart from perhaps T [1 , . ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 15
For a contradiction, suppose that T [1 , . is isomorphic to T [1 , and let φ : T [1 , . → T [1 , be an isomorphism. As φ is order-preserving, we havethat φ (1) = 1 and φ (2 .
5) = 3. Similarly, as φ preserves the semiring multi-plication, we can conclude that φ (2) = φ (1) ⊗ φ (1) = 2 and φ (1 . · φ (1 . ⊗ φ (1) = φ (2 .
5) = 3and hence φ (1 . ≥ φ (2) contradicting that φ is order-preserving. Hence, T [1 , and T [1 , . are not isomorphic.Finally, suppose z > y ≥ φ : T [1 ,y ] → T [1 ,z ] . From the fact that φ is a morphismand the definition of multiplication in the two semirings, we have φ ( a + b ) = φ ( a )+ φ ( b ) for all a, b with a + b ≤ y , and φ (1) = 1. Hence, φ (2) = φ (1+1) = φ (1) + φ (1) = 2, and for 1 ≤ x ≤ y − φ ( x ) = φ ( x + 1 −
1) = φ ( x + 1) − φ (1) = φ (cid:18) x + 12 (cid:19) + φ (cid:18) x + 12 (cid:19) −
1A simple inductive argument using this fact shows that φ (1+ 2 − n ) = 1+ 2 − n for all n ∈ N ∪{ } . Indeed, the base case is the fact that φ (2) = 2, while if theclaim holds for some n then taking x = 1 + 2 − n we have x +12 = 1 + 2 − ( n +1) .Hence by the above φ (1 + 2 − n ) = 2 φ (1 + 2 − ( n +1) ) −
1, so φ (1 + 2 − ( n +1) ) = ( φ (1 + 2 − n ) + 1) = (1 + 2 − n + 1) = 1 + 2 − ( n +1) and the claim holds for n + 1.Note that for any a, b with a + b ≤ φ (1+ a ) = 1+ a and φ (1+ b ) = 1+ b then φ (1 + a + b ) = φ (1 + a ) + φ (1 + b ) − φ (1) = 1 + a + b . By anothersimple induction, we deduce that φ fixes all finite sums of negative powersof 2 (in other words, all dyadic rationals) in the interval [1 , φ fixes everything inthe interval [1 , φ preserves the multiplication in T [1 ,y ] and y < z , it preservesall finite sums which sum to y or less. Since every element in [1 , y ] is a finitesum of values in [1 , φ is the identity function on [1 , y ].Since it is surjective, this means that y = z . (cid:3) Next we observe that, as a consequence of our earlier results, there areexamples of such semirings for which matrix semigroups are not permutablein any rank greater than 1:
Corollary 4.2.
The semigroup M n ( T [0 , ) is permutable if and only if n = 1 .Proof. The semigroup M ( T [0 , ) is commutative and therefore strongly per-mutable. For n >
1, it is easy to see that T [0 , has elements of unboundedmultiplicative order (indeed, for any j ∈ N the element 1 /j has order j ), so M n ( T [0 , ) is not strongly permutable by Theorem 3.6. (cid:3) By Theorem 4.1, we can now always take truncated tropical semirings tobe either of the form T [0 , or T [1 ,z ] . Corollary 4.2 gives a full description ofwhen the matrix semigroups M n ( T [0 , ) are permutable, so we now focus onmatrix semigroups of form M n ( T [1 ,z ] ) for some z > Theorem 4.3. M n ( T [1 , ) is strongly permutable for all n ∈ N . Proof.
We shall show that T [1 , satisfies the hypothesis of Theorem 3.9. Let X = { x , . . . , x k } be a finite subset of T [1 , and X ′ = X ∪ { , −∞} . Definea binary relation ≡ on T [1 , by a ≡ b if and only if a and b either (i) areequal or (ii) are not in X ′ and lie above exactly the same elements of X ′ . Itis easy to see that ≡ is an equivalence relation with at most 2 | X | + 3 classes,in which each element of X occupies its own equivalence class.We must now show that ≡ is a congruence. As T [1 , is commutative, weonly have to show that ≡ is a left congruence. Let x ≡ y . Clearly, if a = 0or −∞ , we have that a ⊗ x ≡ a ⊗ y and a ⊕ x ≡ a ⊕ y . Moreover, if x = y ,we have that a ⊗ x ≡ a ⊗ y and a ⊕ x ≡ a ⊕ y . Hence, as 0 , −∞ ∈ X ′ , wecan assume that a, x, y ≥
1, and thus a ⊗ x = 2 = a ⊗ y .Further, if a ≥ x, y or a ≤ x, y , then clearly a ⊕ x ≡ a ⊕ y . On the otherhand, if a lies between x and y in the order then a , x , y , a ⊕ x and a ⊕ y alllie above exactly the same elements of X ′ , giving that a ⊕ x ≡ a ⊕ y . Thuswe conclude that ≡ is a congruence.Hence, by the usual first isomorphism theorem, the natural morphism T [1 , → T [1 , / ≡ satisfies the conditions of Theorem 3.9, and M n ( T [1 , ) isstrongly permutable for all n ∈ N . (cid:3) The rest of this section treats the remaining truncated tropical semirings,that is, those of the form T [1 ,z ] with z >
2. These will give examples ofsemirings S such that M ( S ) is strongly permutable, but M n ( S ) is notstrongly permutable for all n >
2. We use the notation ⌈ z ⌉ to denotethe smallest integer greater than or equal to z ∈ R . We shall say thata semigroup S is k -permutable if for every s , . . . , s k ∈ S there exists anon-trivial permutation σ ∈ S k such that s σ (1) s σ (2) · · · s σ ( k ) = s s · · · s k . Lemma 4.4.
For z > , let S and S ′ be subsemigroups of M ( T [1 ,z ] ) givenby S = (cid:26)(cid:18) a −∞ b (cid:19) : a, b ∈ T [1 ,z ] (cid:27) and S ′ = (cid:26)(cid:18) −∞ a b (cid:19) : a, b ∈ T [1 ,z ] (cid:27) . Then S and S ′ are both (2 ⌈ z ⌉ + 5) -permutable.Proof. Transposing matrices is a semigroup anti-isomorphism between S and S ′ , so it suffices to prove that S is (2 ⌈ z ⌉ + 5)-permutable.Let m = 2 ⌈ z ⌉ + 5 and let X , . . . , X m ∈ S . If ( X t ) , = −∞ for any t > X t is a right zero of S , we have that X X · · · X m = X X · · · X m .Thus we may assume ( X t ) , = −∞ for all t > X t ) , , ( X t +1 ) , = −∞ for some t < m then as diagonal matricescommute, we have X · · · X t X t +1 · · · X m = X · · · X t +1 X t · · · X m . Therefore,we may assume either ( X ) , = −∞ or ( X ) , = −∞ . Combined with theassumption from the previous paragraph, this implies we may assume that( X · · · X m ) , = −∞ .If ( X t ) , , ( X t +1 ) , = 0 for some t < m then, because 2 × X · · · X t X t +1 · · · X m = X · · · X t +1 X t · · · X m .Hence, we may assume that among every pair of every two consecutivematrices (except perhaps the first three) there is a matrix X t with ( X t ) , ≥
1. Since m = 2 ⌈ z ⌉ + 5 this means we have ( X · · · X m − ) , = z and( X · · · X m − ) , = z or −∞ . In both of these cases X · · · X m − acts as a ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 17 left zero for all matrices M with M , = −∞ . But we assumed ( X t ) , = −∞ for t >
2, so we have X · · · X m = X · · · X m − X m − X m = X · · · X m − X m X m − . Thus S , and hence also S ′ , is (2 ⌈ z ⌉ + 5)-permutable. (cid:3) Lemma 4.5.
Let A ∈ M ( T [1 ,z ] ) and m be the minimum finite entry of A (or m = z if A if all entries are −∞ ). Let k ≥ ⌈ z ⌉ + 45) . Then forall A , . . . , A k ∈ M ( T [1 ,z ] ) , either ( A A · · · A k ) i,j = m for all i, j or there exists a non-trivial σ ∈ S k such that A A A · · · A k = A A σ (1) A σ (2) · · · A σ ( k ) Proof.
Consider a product A A . . . A k . If the product does not contain an m we are done. Moreover, as M ( T [ x,z ] \ { } ) is an ideal of M ( T [1 ,z ] ) for all x ∈ [1 , z ], we may suppose every truncated product A A . . . A p with p < k has at least one entry equal to m .By the pigeon hole principle there exists a sequence of indices 0 ≤ i < · · · < i n ≤ k where n = (cid:6) k (cid:7) − A A · · · A i j has m in the same position. If this is the (1,2) or the (2,1) position then notethat swapping the rows of A swaps the rows of the product A A · · · A t for all t ≤ k . Therefore if σ is a permutation that does not change theproduct, then σ will also preserve the product obtained by swapping A ’srows. Hence, we can assume that the m ’s are in the (1,1) or (2,2) position.Moreover, by relabelling the rows and columns if necessary, we can assumewithout loss of generality that A A · · · A i j has m in the (1,1) position forall 0 ≤ j ≤ n .Now consider the matrices defined by B = A · · · A i and B j = A i j − +1 · · · A i j for 1 ≤ j ≤ n . Any permutation of this sequence which does not change theproduct clearly yields a permutation of the original sequence which does notchange the product, so it is enough to seek a non-trivial permutation of thissequence. We define the truncated products Σ t := BB · · · B t for 0 ≤ t ≤ n .First we consider any matrices B i whose entries are all either 0 or −∞ .There are only 16 distinct matrices of this form, so if more than 16 ofthe B i s have this form then the same matrix would appear twice in thesequence, resulting in a non-trivial permutation that preserves the product.Otherwise, since n = (cid:6) k (cid:7) − > ⌈ z ⌉ + 11) the B i s contain a subsequenceof 4 ⌈ z ⌉ + 11 consecutive matrices not of this form, say B p , . . . , B q where q − p = 4 ⌈ z ⌉ + 10.We now define five subsets of M ( T [1 ,z ] ): S = (cid:26)(cid:18) a −∞ b (cid:19) : a, b ∈ T [1 ,z ] (cid:27) , S ′ = (cid:26)(cid:18) −∞ a b (cid:19) : a, b ∈ T [1 ,z ] (cid:27) ,T = (cid:26)(cid:18) a b (cid:19) : a, b ∈ T [1 ,z ] (cid:27) , U = (cid:26)(cid:18) −∞ a b (cid:19) : a, b ∈ T [1 ,z ] (cid:27) ,V = (cid:26)(cid:18) ca b (cid:19) : a, b, c ∈ T [1 ,z ] (cid:27) . We shall show that the sequence of B i s contains 2 ⌈ z ⌉ +5 consecutive matriceseither all in S or all in S ′ . From this it will follow by Lemma 4.4 that thereis a permutation of the sequence which preserves the product, as required.Note that S, S ′ , T ⊆ V . For p ≤ t ≤ q −
1, we have that (Σ t ) , =(Σ t +1 ) , = m . So, if (Σ t ) , = −∞ , then in order to ensure (Σ t B t +1 ) , =(Σ t +1 ) , = m we must have ( B t +1 ) , = 0, that is, B t +1 ∈ V . Similarly, if(Σ t ) , = m , then B t +1 ∈ S, T or U . Otherwise, (Σ t ) , > m and we havethat B t +1 ∈ S .If the matrices B p , . . . , B p +2 ⌈ z ⌉ +4 are all in S ′ then we are done. Oth-erwise, choose t with p ≤ t ≤ p + 2 ⌈ z ⌉ + 4 such that B t / ∈ S ′ . Since(Σ t − ) , = m and Σ t = Σ t − B t , this means that (Σ t ) , = −∞ .Now because (Σ t ) , (Σ t ) , ≥ m and B t +1 lies in S , T or U with (becauseof the assumption that the entries of B t +1 are not all 0 and −∞ ) either( B t +1 ) , ≥ B t +1 ) , ≥
1, we have that (Σ t +1 ) , > m and of courseby definition we have (Σ t +1 ) , ≥ m . Continuing by induction we deducethat (Σ i ) , > m for all i with t + 1 ≤ i ≤ q . By the remarks in the lastparagraph but one, this means that B j ∈ S for all t + 2 ≤ j ≤ q , whichmeans the matrices B t +2 , . . . B t +1+2 ⌈ z ⌉ +5 are all in S , as required. (cid:3) Theorem 4.6.
Let z > . Then M ( T [1 ,z ] ) is strongly permutable.Proof. Consider a product of matrices A · · · A n for n ≥ ⌈ z ⌉ +1)(16 ⌈ z ⌉ +45) and let m t be the smallest finite entry in the product of the first t matrices Σ t = A · · · A t . (If all entries of Σ t are −∞ , we define m t = z ). Notethat m ≤ · · · ≤ m n as M ( T [ x,z ] \ { } ) is an ideal of M ( T [1 ,z ] ) for all x ∈ [1 , z ]. Further, let k , . . . , k s be all the values such that m k j − < m k j . For acontradiction, suppose that there does not exist a non-trivial permutation σ ∈ S n such that A · · · A n = A σ (1) · · · A σ ( n ) . Then, by Lemma 4.5, we havethat s > k j − k j − < ⌈ z ⌉ + 45) for all j as there is nopermutation preserving the product A · · · A n by assumption.For any 1 ≤ j ≤ s −
4, consider the five values m k j < m k j +1 < m k j +2
Let z > . Then M n ( T [1 ,z ] ) is strongly permutable if andonly if n < .Proof. The direct implication is Theorem 4.6. For the converse implicationit suffices by Proposition 3.1 to show that M ( T [1 ,z ] ) is not permutable. Wedo this by a variation of the method used to prove Lemma 3.3 above.Choose ε with 0 < ε < z −
2. For a fixed m , we define a sequence ofmatrices B , B , . . . , B m by B i = im ε ε −∞ ε − i − m ε −∞ −∞ ERMUTABILITY OF MATRICES OVER BIPOTENT SEMIRINGS 19
By induction the product of the first k such matrices is given by k Y i =1 B i = km ε ε −∞ ε −∞ −∞ Now, suppose σ ∈ S m is such that B σ := Q mi =1 B σ ( i ) = Q mi =1 B i . By thedefinition of matrix multiplication, for any j < k we must have2 + ε = ( B σ ) , ≥ ( B σ ( j ) ) , + ( B σ ( k ) ) , = 2 + ε + εm ( σ ( j ) − σ ( k ) + 1)and hence σ ( j ) < σ ( k ). Since σ is a permutation this means σ is the identitypermutation. Further, as m was arbitrary M ( T [1 ,z ] ) is not permutable, sotogether with the previous theorem we get that M n ( T [1 ,z ] ) is permutable ifand only if n < (cid:3) References [1] C. Brackley, D. Broomhead, M. Romano and M. Thiel, A max-plus model of ribosomedynamics during mRNA translation, Journal of Theoretical Biology, 303 (2012) 128–140.[2] G. Cohen, S. Gaubert and J. Quadrat, Max-plus algebra and system theory: Wherewe are and where to go now, Annual Reviews in Control, 23 (1999) 207–219.[3] F. d’Alessandro and E. Pasku, A combinatorial property for semigroups of matrices,Semigroup Forum, 67(1), (2003) 22–30.[4] J. Golan, Semirings and their Applications, Springer 1999.[5] Z. Izhakian, Tropical plactic algebra, the cloaktic monoid, and semigroup representa-tions, Journal of Algebra, 524 (2019) 290–366.[6] Z. Izhakian and S. Margolis, Semigroup identities in the monoid of two-by-two tropicalmatrices. Semigroup Forum, 80(2) (2010) 191–218.[7] M. Johnson and M. Kambites, Tropical representations and identities of placticmonoids, Trans. Amer. Math. Soc. (to appear).[8] J. Kirby, An Invitation to Model Theory, Cambridge University Press 2019.[9] I. Klimann, S. Lombardy, J. Mairesse and C. Prieur, Deciding unambiguity and se-quentiality from a finitely ambiguous max-plus automaton, Theoretical Computer Sci-ence, 327(3) (2004) 349–373.[10] G. Mikhalkin, Enumerative tropical algebraic geometry in R , J. Amer. Math. Soc.,18(2) (2005) 313–377.[11] J. Okni´nski, Semigroup Algebras, Marcel Dekker 1991.[12] M. Taylor, On upper triangular tropical matrix semigroups, tropical matrix identitiesand T -modules-modules