Algebraic structures of tropical mathematics
aa r X i v : . [ m a t h . R A ] M a y ALGEBRAIC STRUCTURES OF TROPICAL MATHEMATICS
ZUR IZHAKIAN, MANFRED KNEBUSCH, AND LOUIS ROWEN
Abstract.
Tropical mathematics often is defined over an ordered cancellative monoid M , usually taken tobe ( R , +) or ( Q , +). Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semiringspossess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties,thereby forcing researchers to rely often on combinatoric techniques.In this paper we describe an alternative structure, more compatible with valuation theory, studied bythe authors over the past few years, that permits fuller use of algebraic theory especially in understandingthe underlying tropical geometry. The idempotent max-plus algebra A of an ordered monoid M is replacedby R := L ×M , where L is a given indexing semiring (not necessarily with 0). In this case we say R layered by L . When L is trivial, i.e, L = { } , R is the usual bipotent max-plus algebra. When L = { , ∞} werecover the “standard” supertropical structure with its “ghost” layer. When L = N we can describe multipleroots of polynomials via a “layering function” s : R → L .Likewise, one can define the layering s : R ( n ) → L ( n ) componentwise; vectors v , . . . , v m are called tropically dependent if each component of some nontrivial linear combination P α i v i is a ghost, for“tangible” α i ∈ R . Then an n × n matrix has tropically dependent rows iff its permanent is a ghost.We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraicfoundation for tropical linear algebra, in which many classical tools are available. In the process, we providesome new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework ofidentities of semirings. Introduction
Tropical geometry, a rapidly growing area expounded for example in [Gat, ItMS, L1, MS, SS], has beenbased on two main approaches. The most direct passage to tropical mathematics is via logarithms. Butvaluation theory has richer algebraic applications (for example providing a quick proof of Kapranov’s theo-rem), and much of tropical geometry is based on valuations on Puiseux series. The structures listed aboveare compatible with valuations, and in § • Algebraic geometry as espoused by Zariski and Grothendieck, using varieties and commutative al-gebra in the context of category theory. • Linear algebra via tropical dependence, the characteristic polynomial, and (generalized) eigenspaces. • Algebraic formulations for more sophisticated concepts such as resultants, discriminants, and Jaco-bians.
Date : November 6, 2017.2010
Mathematics Subject Classification.
Primary 06F20, 11C08, 12K10, 14T05, 14T99, 16Y60; Secondary 06F25, 16D25.
Key words and phrases. tropical algebra, layered supertropical domains, polynomial semiring, d-base, s-base, bilinear form.The research of the first and third authors was supported by the Israel Science Foundation (grant No. 448/09).The research of the first author also was conducted under the auspices of the Oberwolfach Leibniz Fellows Programme(OWLF), Mathematisches Forschungsinstitut Oberwolfach, Germany.
This approach leads to the use of polynomials and matrices, which requires two operations. Our taskhas been to pinpoint the appropriate category of semirings in which to work, or equivalently, how far do wedequantize in the process of tropicalization? In this survey we compare four structures, listed in increasinglevel of refinement: • The max-plus algebra, • Supertropical algebra, • Layered tropical algebras, • Exploded supertropical algebras.We review the layered algebra in §
3, compare it to the max-plus algebra, and then in § § Algebraic Background
We start by reviewing some notions which may be familiar, but are needed extensively in our exposition.The basic tropicalization, or dequantization, involves taking logarithms to ( R , + ), which as explained in [L1]replaces conventional multiplication by addition, and conventional addition by the maximum. This is calledthe max-plus algebra of ( R , + ).2.1. Ordered groups and monoids.
Recall that a monoid ( M , · ,
1) is a set with an associative operation · and a unit element 1. We usually work with Abelian monoids, in which the operation is commutative. Thepassage to the max-plus algebra in tropical mathematics can be viewed algebraically via ordered groups(such as ( R , + )), and, more generally, ordered monoids.An Abelian monoid M := ( M , · ,
1) is cancellative if ab = ac implies b = c. There is a well-knownlocalization procedure with respect to a submonoid S of a cancellative Abelian monoid M , obtained bytaking M × S/ ∼ , where ∼ is the equivalence relation given by ( a, s ) ∼ ( a ′ , s ′ ) iff as ′ = a ′ s. Localizingwith respect to all of M yields its group of fractions , cf. [Bo, W]. We say that a monoid M is power-cancellative (called torsion-free by [W]) if a n = b n for some n ∈ N implies a = b. A monoid M is called N - divisible (also called radicalizible in the tropical literature) if for each a ∈ M and m ∈ N there is b ∈ M such that b m = a. For example, ( Q , + ) is N -divisible. Remark 2.1.
The customary way of embedding an Abelian monoid M into an N -divisible monoid, is toadjoin m √ a for each a ∈ M and m ∈ N , and define m √ a n √ b := mn √ a n b m . This will be power-cancellative if M is power-cancellative. An ordered Abelian monoid is an Abelian monoid endowed with a total order satisfying the property: a ≤ b implies ga ≤ gb, (2.1)for all elements a, b, g . Any ordered cancellative Abelian monoid is infinite.One advantage of working with ordered monoids and groups is that their elementary theory is well-knownto model theorists. The theory of ordered N -divisible Abelian groups is model complete, cf. [M, p. 116]and [Sa, pp. 35, 36], which essentially means that every N -divisible ordered cancellative Abelian monoid hasthe same algebraic theory as the max-plus algebra ( Q , + ), which is a much simpler structure than ( R , + ).From this point of view, the algebraic essence of tropical mathematics boils down to ( Q , + ). Sometimes wewant to study its ordered submonoid ( Z , + ), or even ( N , + ), although they are not N -divisible.Nevertheless, just as one often wants to study the arithmetic of Q by viewing finite homomorphic imagesof Z , we want the option of studying finite homomorphic images of the ordered monoid ( N , + ). Towardsthis end, we define the q - truncated monoid M = [1 , q ] := { , , . . . , q } , given with the obvious ordering;the sum and product of two elements k, ℓ ∈ L are taken as usual, if not exceeding q − , and is q otherwise.In other words, q could be considered as the infinite element of the finite monoid M . LGEBRAIC STRUCTURES 3
Semirings without zero.
So far, dequantization has enabled us to pass from algebras to orderedAbelian monoids, which come equipped with a rich model theory ready to implement, and as noted above,there is a growing theory of algebraic geometry over monoids [CHWW]. But to utilize standard tools such aspolynomials and matrices, we need two operations (addition and multiplication), and return to the languageof semirings, using [Gol] as a general reference. We write † to indicate that we do not require the zeroelement.A semiring † ( R, + , · ,
1) is a set R equipped with binary operations + and · such that: • ( R, +) is an Abelian semigroup; • ( R, · , R ) is a monoid with identity element R ; • Multiplication distributes over addition.A semifield † is a semiring † in which every element is (multiplicatively) invertible. In particular, themax-plus algebras ( Z , + ), ( Q , + ), and ( R , + ) are semifields † , since + now is the multiplication.A semiring is a semiring † with a zero element R satisfying a + R = a, a · R = R = R · a, ∀ a ∈ R. We use semirings † instead of semirings since the zero element can be adjoined formally, and often isirrelevant. For example, the zero element of the max-plus algebra would be −∞ , which requires specialattention.A semifield is a semifield † with a zero element adjoined. Note that under this definition the customaryfield Q with the usual operations is not a semifield, since Q \ { } is not closed under addition.Any ordered Abelian monoid gives rise to a max-plus semiring † , where the operations are written ⊙ and ⊕ and defined by: a ⊕ b := max { a, b } ; a ⊙ b := a + b. Associativity and distributivity (of ⊙ over ⊕ ) hold, but NOT negation, since a ⊕ b = −∞ unless a = b = −∞ . Although the circle notation is standard in the tropical literature, we find it difficult to read whendealing with algebraic formulae. (Compare x + 7 x + 4 x + 1 with x ⊙ x ⊙ x ⊙ x ⊕ ⊙ x ⊙ x ⊙ x ⊕ ⊙ x ⊕ . )Thus, when appealing to the abstract theory of semirings we use the usual algebraic notation of · (oftensuppressed) and + respectively for multiplication and addition.The max-plus algebra satisfies the property that a + b ∈ { a, b } ; we call this property bipotence . Inparticular, the max-plus algebra, viewed as a semiring † , is idempotent in the sense that a + a = a for all a .Although idempotence pervades the theory, it turns out that what is really crucial for many applications isthe following fact: Remark 2.2.
In any idempotent semiring † , if a + b + c = a , then a + b = a. (Proof: a = a + b + c =( a + b + c ) + b = a + b .) Let us call such a semiring † proper . Note that a proper semiring cannot have additive inverses otherthan , since if c + a = , then a = a + = a + c + a, implying a = a + c = . Any proper semiring † R gives rise to a partial order, given by a ≤ b iff a + c = b for some c ∈ R . Thisis a total order when the semiring † R is bipotent. Thus, the categories of bipotent semirings † and orderedmonoids are isomorphic, and each language has its particular advantages.2.3. The function semiring † .Definition 2.3. The function semiring † Fun(
S, R ) is the set of functions from a set S to a semiring † R . Fun(
S, R ) becomes a semiring † under componentwise operations, and is proper when R is proper. Cus-tomarily one takes S = R ( n ) , the Cartesian product of n copies of R . This definition enables us to workwith proper subsets, but the geometric applications lie outside the scope of the present paper.2.3.1. Polynomials and power series.
Λ = { λ , . . . , λ n } always denotes a finite set of indeterminates com-muting with the semiring † R ; often n = 1 and we have a single indeterminate λ . We have the polynomialsemiring † R [Λ]. As in [IzR1], we view polynomials in R [Λ] as functions, but perhaps viewed over someextension R ′ of R . More precisely, for any subset S ⊆ R ( n ) , there is a natural semiring † homomorphism ψ : R [Λ] → Fun(
S, R ) , (2.2) Z. IZHAKIAN, M. KNEBUSCH, AND L. ROWEN obtained by viewing a polynomial as a function on S .When R is a semifield † , the same analysis is applicable to Laurent polynomials R [Λ , Λ − ], since thehomomorphism λ i a i then sends λ − i a − i . Likewise, when R is power-cancellative and divisible, wecan also define the semiring † of rational polynomials R [Λ] rat , where the powers of the λ i are taken to bearbitrary rational numbers. These can all be viewed as elementary formulas in the appropriate languages,so the model theory alluded to earlier is applicable to the appropriate polynomials and their (tropical) rootsin each case.Other functions over the bipotent semiring † R of an ordered monoid M can be defined in the same way.For example, if M is an ordered submonoid of ( R + , · ) , then we can define the formal exponential seriesexp( a ) := X k a k k ! (2.3)since a < m implies a m +1 ( m +1)! < a m m ! , and thus (2.3) becomes a finite sum. It follows at once that exp( λ ) := P λ k k ! is defined in Fun( R, R ) . Puisuex series and valuations.
Since logarithms often do not work well with algebraic structure ,tropicalists have turned to the algebra of Puiseux series , denoted K , whose elements have the form p ( t ) = X τ ∈ Q ≥ , c τ ∈ K c τ t τ , where the powers of t are taken over well-ordered subsets of Q . Here K is any algebraically closed field ofcharacteristic 0, customarily C . Intuitively, we view t as a “generic element.” In the literature, the powers τ are often taken in R rather than Q , but it is enough to work with Q , for which it much easier to computethe powers of t . The algebra K is an algebraically closed field.Now recall that a valuation from an integral domain W to an ordered monoid ( G , + ) is a multiplicativemonoid homomorphism v : W \ { } → G , i.e., with v ( ab ) = v ( a ) + v ( b ) , and satisfying the property v ( a + b ) ≥ min { v ( a ) , v ( b ) } for all a, b ∈ K. We formally put v (0) = ∞ . Forexample, the field of Puiseux series has the order valuation v given by v ( p ( t )) := min { τ ∈ Q ≥ : c τ = 0 } . As t → , the dominant term in p ( t ) becomes c v ( p ( t )) t v ( p ( t )) . The following basic observation in valuation theory shows why valuations are relevant to the tropicaltheory.
Remark 2.4. If v ( a ) = v ( b ) , then v ( a + b ) = min { v ( a ) , v ( b ) } . Inductively, if v ( a ) , . . . , v ( a m ) are distinct,then v (cid:18) m X i =1 a i (cid:19) = min { v ( a i ) : 1 ≤ i ≤ m } ∈ G . Consequently, if P a i = 0 , then at least two of the v ( a i ) are the same. These considerations are taken muchmore deeply in [BiG] . When W is a field, the value monoid G is a group. Much information about a valuation v : W → G ∪ {∞} can be garnered from the target v ( W ), but valuation theory provides some extra structure: • The valuation ring O v = { a ∈ W : v ( a ) ≥ } , • The valuation ideal P v = { a ∈ W : v ( a ) > } , • The residue ring ¯ W = O v /P v , a field if W is a field.For example, the valuation ring of the order valuation on the field K of Puiseux series is { p ( t ) ∈ K : c τ = 0for τ < } , and the residue field is K .We replace v by − v to switch minimum to maximum, and ∞ by −∞ . One can generalize the notion ofvaluation to permit W to be a semiring † ; taking W = M , we see that the identity map is a valuation, whichprovides one of our main examples. LGEBRAIC STRUCTURES 5
The standard supertropical semiring † . This construction, following [IzR1], refines the max-plusalgebra and picks up the essence of the value monoid. From now on, in the spirit of max-plus, we write theoperation of an ordered monoid M as multiplication.We start with an Abelian monoid M := ( M , · ), an ordered group G := ( G , · ), and an onto monoidhomomorphism v : M → G . We write a ν for v ( a ) , for a ∈ M . Thus every element of G is some a ν . We write a ∼ = ν b if a ν = b ν . Our two main examples: • M = G is the ordered monoid of the max-plus algebra (the original example in Izhakian’s disserta-tion); • M is the multiplicative group of a field F , and v : F × → G is a valuation. Note that we forget theoriginal addition on the field F !Our objective is to use the order on G to study M . Accordingly we want to define a structure on M ∪ G .The standard supertropical semiring † R is the disjoint union M ∪ G , made into a monoid by startingwith the given multiplications on M and G , and defining a · b ν and a ν · b to be ( ab ) ν for a, b ∈ M . Weextend v to the ghost map ν : R → G by taking ν | M = v and ν | G to be the identity on G . Thus, ν is amonoid projection.We make R into a semiring † by defining a + b = a for a ν > b ν ; b for a ν < b ν ; a ν for a ν = b ν .R is never additively cancellative (except for M = { } ). M is called the tangible submonoid of R . G is called the ghost ideal . R is called a supertropical domain † when the monoid M is (multiplicatively) cancellative.Strictly speaking, a supertropical domain † will not be a semifield † since the ghost elements are notinvertible. Accordingly, we define a † to be a supertropical domain † for which M is a group.Motivation: The ghost ideal G is to be treated much the same way that one treats the zero element incommutative algebra. Towards this end, we write a | gs = b if a = b or a = b + ghost . (Accordingly, write a | gs = if a is a ghost.) Note that for a tangible, a | gs = b iff a = b. If needed, we couldformally adjoin a zero element in a separate component; then the ghost ideal is G := G ∪ { } . We maythink of the ghost elements as uncertainties in classical algebra arising from adding two Puiseux series whoselowest order terms have the same degree. R is a cover of the max-plus algebra of G , in which we “resolve” tangible idempotence, in the sense that a + a = a ν instead of a + a = a. This modification in the structure permits us to detect corner roots of tropical polynomials in terms ofthe algebraic structure, by means of ghosts. Namely, we say that a ∈ R ( n ) is a root of a polynomial f ∈ R [Λ]when f ( a ) ∈ G . This concise formulation enables us to apply directly many standard mathematical conceptsfrom algebra, algebraic geometry, category theory, and model theory, as described in [IzKR1]–[IzKR5] and[IzR1]–[IzR6].The standard supertropical semiring works well with linear algebra, as we shall see.2.6.
Kapranov’s Theorem and the exploded supertropical structure.
Given a polynomial f (Λ) = P i p i ( λ i · · · λ i n n ) ∈ K [Λ] , where i = ( i , . . . , i n ), i.e., with each p i a Puiseux series, we define its tropicaliza-tion ˜ f to be the tropical polynomial P i v ( p i ) λ i · · · λ i n n . (In the tropical literature, this is customarily writtenin the circle notation.) By Remark 2.4, if a ∈ K ( n ) is a root of f in the classical sense, then v ( a ) is a tropicalroot of ˜ f . Kapranov showed, conversely, that any tropical root of ˜ f has the form v ( a ) for suitable a ∈ K ( n ) , and valuation theory can be applied to give a rather quick proof of this fact, although we are not aware ofan explicit reference. (See [R1, Proposition 12.58] for an analogous proof of a related valuation-theoreticresult.)To prove Kapranov’s theorem, one needs more than just the lowest powers of the Puiseux series appearingas coefficients of f , but also their coefficients; i.e., we also must take into account the residue field of the Z. IZHAKIAN, M. KNEBUSCH, AND L. ROWEN order valuation on Puiseux series. Thus, we need to enrich the supertropical structure to include this extrainformation. This idea was first utilized by Parker [Par] in his “exploded” tropical mathematics. Likewise,Kapranov’s Theorem has been extended by Payne [Pay1, Pay2], for which we need the following more refinedsupertropical structure, initiated by Sheiner [ShSh]:
Definition 2.5.
Given a valuation v : W → G , we define the exploded supertropical algebra R = W × G , viewed naturally as a monoid. (Thus we are mixing the “usual” world with the tropical world.)We make R into a semiring † by defining ( c, a ) + ( d, b ) = ( c, a ) when a > b ;( d, b ) when a < b ;( c + d, a ) when a = b. Sheiner’s theory parallels the standard supertropical theory, where now the ghost elements are taken tobe the 0-layer { } × G . The layered structure
The standard supertropical theory has several drawbacks. First, it fails to detect the multiplicity of aroot of a polynomial. For example we would want 3 to have multiplicity 5 as a tropical root of the tropicalpolynomial ( λ + 3) ; this is not indicated supertropically. Furthermore, serious difficulties are encounteredwhen attempting to establish a useful intrinsic differential calculus on the supertropical structure. Also,some basic supertropical verifications require ad hoc arguments.These drawbacks are resolved by refining the ghost ideal into different “layers,” following a constructionof [WW, Example 3.4] and [AkGG, Proposition 5.1]. Rather than a single ghost layer, we take an indexingset L which itself is a partially ordered semiring † ; often L = N under classical addition and multiplication.Ordered semirings † can be trickier than ordered groups, since, for example, a > b in ( R , · ) does not imply − a > − b, but rather − a < − b. To circumvent this issue, we require all elements in the indexing semiring † to be non-negative. Construction 3.1 ( [IzKR4, Construction 3.2]) . Suppose we are given a cancellative ordered monoid G ,viewed as a semiring † as above. For any partially ordered semiring † L we define the semiring † R := R ( L, G ) to be set-theoretically L × G , where we denote the “layer” { ℓ } × G as R ℓ and the element ( ℓ, a ) as [ ℓ ] a ; wedefine multiplication componentwise, i.e., for k, ℓ ∈ L, a, b ∈ G , [ k ] a [ ℓ ] b = [ kℓ ] ( ab ) , (3.1) and addition via the rules: [ k ] a + [ ℓ ] b = [ k ] a if a > b, [ ℓ ] b if a < b, [ k + ℓ ] a if a = b. (3.2) The sort map s : R → L is given by s ( [ k ] a ) = k . R is indeed a semiring † . We identify a ∈ G with [1] a ∈ R .In most applications the “sorting” semiring † L is ordered, and its smallest nonzero element is 1. In thiscase, the monoid { [ ℓ ] a : 0 < ℓ ≤ } is called the tangible part of R . The ghosts are { [ ℓ ] a : ℓ > } , andcorrespond to the ghosts in the standard supertropical theory. The ghosts together with R comprise anideal. If there is a zero element it would be [0] .One can view the various choices of the sorting semiring † L as different stages of degeneration of algebraicgeometry, where the crudest (for L = { } ) is obtained by passing directly to the familiar max-plus algebra.The supertropical structure is obtained when L = { , ∞} , where R and R ∞ are two copies of G , with R the tangible submonoid of R and R ∞ being the ghost copy. Other useful choices of L include { , , ∞} (todistinguish between simple roots and multiple roots) and N , which enables us to work with the multiplicityof roots and with derivatives, as seen below. In order to deal with tropical integration as anti-differentiation,one should consider the sorting semirings † Q > and R > , but this is outside our present scope.By convention, [ ℓ ] λ denotes [ ℓ ] R λ. Thus, any monomial can be written in the form [ ℓ ] α i λ i · · · λ i n n where i = ( i , . . . , i n ) . We say a polynomial f is tangible if each of its coefficients is tangible. LGEBRAIC STRUCTURES 7
Note that the customary decomposition R = L ℓ ∈ L R ℓ in graded algebras has been strengthened to thepartition R = ˙ S ℓ ∈ L R ℓ . The ghost layers now indicate the number of monomials defining a corner root of atangible polynomial. Thus, we can measure multiplicity of roots by means of layers. For example,( λ + 3) = [1] λ + [5] λ + [10] λ + [10] λ + [5] λ + [1] , and substituting 3 for λ gives [32]
15 = [2 ] . Layered derivatives.
Formal derivatives are not very enlightening over the max-plus algebra. Forexample, if we take the polynomial f = λ + 5 λ + 8, which has corner roots 3 and 5, we have f ′ = 2 λ + 5 , having corner root 3, but the common corner root 3 of f and f ′ could hardly be considered a multiple rootof f . This difficulty arises from the fact that 1 + 1 = 2 in the max-plus algebra. The layering permits us todefine a more useful version of the derivative (where now R contains a zero element R ): Definition 3.2.
The layered derivative f ′ lay of f on R [ λ ] is given by: (cid:18) n X j =0 [ ℓ j ] α j λ j (cid:19) ′ lay := n X j =1 [ jℓ j ] α j λ j − . (3.3)In particular, for α = [1] α ∈ R , ( αλ j ) ′ lay := [ j ] α λ j − ( j ≥ , ( αλ ) ′ lay := α, and α ′ lay := R . Thus, we have the familiar formulas:(1) ( f + g ) ′ lay = f ′ lay + g ′ lay ;(2) ( f g ) ′ lay = f ′ lay g + f g ′ lay .This is far more informative in the layered setting (say for L = N ) than in the standard supertropicalsetting, in which ( αλ j ) ′ is ghost for all j ≥ The tropical Laplace transform.
The classical technique of Laplace transforms has a tropical analogwhich enables us to compare the various notions of derivative. Suppose L is infinite, say L = N . Formallypermitting infinite vectors ( a ℓ ) ℓ ∈ L permits us to define a homomorphism R [[Λ]] → R ( L, R ) given by X a k λ k (cid:0) [ k ] k ! a k (cid:1) . (Strictly speaking, we would want the image to be ( [ k ] k ! a k ) , but this would complicate the notation andrequire us to take L = Q + .) For example, exp lay ( a ) ( [ k ] a k ) where each a k = a. Now we define ( [ ℓ ] a ℓ ) ′ = ( [ ℓ − a ℓ ) . Then exp ′ lay = exp lay . This enables one to handle trigonometricfunctions in the layered theory.3.3.
Layered domains † with symmetry, and patchworking. Akian, Gaubert, and Guterman [AkGG,Definition 4.1] introduced an involutory operation on semirings, which they call a symmetry , to unify thesupertropical theory with classical ring theory. One can put their symmetry in the context of R ( L, G ). Definition 3.3. A negation map on a semiring † L is a function τ : L → L satisfying the properties: N1. τ ( kℓ ) = τ ( k ) ℓ = kτ ( ℓ ) ; N2. τ ( k ) = k ; N3. τ ( k + ℓ ) = τ ( k ) + τ ( ℓ ) . Suppose the semiring † L has a negation map τ of order ≤
2. We say that R := R ( L, G ) has a symmetry σ when R is endowed with a map σ : R → R and a negation map τ on L , together with the extra axiom:S1. s ( σ ( a )) = τ ( s ( a )) , ∀ a ∈ R. Z. IZHAKIAN, M. KNEBUSCH, AND L. ROWEN
Example 3.4.
Suppose L is an ordered semiring † . We mimic the well-known construction of Z from N . Define the doubled semiring † D ( L ) = L × L − , the direct product of two copies L and L − of L , where addition is defined componentwise, but multiplicationis given by ( k, ℓ ) · ( k ′ , ℓ ′ ) = ( kk ′ + ℓℓ ′ , kℓ ′ + ℓk ′ ) . In other words, D ( L ) is multiplicatively graded by {± } .D ( L ) is endowed with the product partial order, i.e., ( k ′ , ℓ ′ ) ≥ ( k, ℓ ) when k ′ ≥ k and ℓ ′ ≥ ℓ . Here is an example relating to “patchworking,” [ItMS].
Example 3.5.
Suppose G is an ordered Abelian monoid, viewed as a semiring † as in Construction 3.1.Define the doubled layered domain † R = R ( D ( L ) , G ) = { (( k, ℓ ) , a ) : ( k, ℓ ) = (0 , , a ∈ G} , but with addition and multiplication given by the following rules: (( k, ℓ ) , a ) + (( k ′ , ℓ ′ ) , b ) = (( k, ℓ ) , a ) if a > b, (( k ′ , ℓ ′ ) , b ) if a < b, (( k + k ′ , ℓ + ℓ ′ ) , a ) if a = b. (( k, ℓ ) , a ) · (( k ′ , ℓ ′ ) , b ) = (( kk ′ + ℓℓ ′ , kℓ ′ + k ′ ℓ ) , ab ) . Remark 3.6. In R = R ( D ( L ) , G ) , the symmetry σ : R → R given by σ : (( k, ℓ ) , a ) (( ℓ, k ) , a ) is analogousto the one described in [AkGG] , and behaves much like negation. For example, when L = { , ∞} , we note that D ( L ) = { (1 , , (1 , ∞ ) , ( ∞ , , ( ∞ , ∞ ) } , which is applicableto Viro’s theory of patchworking, where the “tangible” part could be viewed as those elements of layer(1 , , (1 , ∞ ) , or ( ∞ , § , , and − ∞ , ∞ ) with the set { , , − } .4. Matrices and linear algebra
As an application, the supertropical and layered structures provide many of the analogs to the classicalHamilton-Cayley-Frobenius theory. M n ( R ) denotes the semiring † of n × n matrices over a semiring R . (Notethat the familiar matrix operations do not require negation.)Although one of the more popular and most applicable aspects of idempotent mathematics, idempotentmatrix theory is handicapped by the lack of an element − The supertropical determinant.
This discussion summarizes [IzR3]. We define the supertropicaldeterminant | A | of a matrix A = ( a i,j ) to be the permanent: | ( a i,j ) | = X π ∈ S n a ,π (1) · · · a n,π ( n ) . (4.1)Defining the transpose matrix ( a i,j ) t to be ( a j,i ) , we have (cid:12)(cid:12) ( a i,j ) t (cid:12)(cid:12) = | ( a i,j ) | . | A | = R iff “enough” entries are R to force each summand in Formula 4.1 to be R . This property,which in classical matrix theory provides a description of singular subspaces, is too strong for our purposes.We now take the natural supertropical version. Write T for the tangible elements of our supertropicalsemiring R , and T = T ∪ { } . Definition 4.1.
A matrix A is nonsingular if | A | ∈ T ; A is singular when | A | ∈ G . LGEBRAIC STRUCTURES 9
The standard supertropical structure often is sufficient for matrices, since it enables us to distinguishbetween nonsingular matrices (in which the tropical n × n determinant is computed as the unique maximalproduct of n elements in one track) and singular matrices.The tropical determinant is not multiplicative, as seen by taking the nonsingular matrix A = (cid:18) (cid:19) .Then A = (cid:18) (cid:19) is singular and (cid:12)(cid:12) A (cid:12)(cid:12) = 5 ν = 2 ·
2. But we do have:
Theorem 4.2.
For any n × n matrices over a supertropical semiring R , we have | AB | | gs = | A | | B | . In particular, | AB | = | A | | B | whenever | AB | is tangible. We say a permutation σ ∈ S n attains | A | if | A | ∼ = ν a σ (1) , · · · a σ ( n ) ,n . • By definition, some permutation always attains | A | . • If there is a unique permutation σ which attains | A | , then | A | = a ,σ (1) · · · a n,σ ( n ) . • If at least two permutations attain | A | , then A must be singular. Note in this case that if we replacedall nonzero entries of A by tangible entries of the same ν -value, then A would still be singular.4.2. Quasi-identities and the adjoint.Definition 4.3. A quasi-identity matrix I G is a nonsingular, multiplicatively idempotent matrix equal to I + Z G , where Z G is R on the diagonal, and whose off-diagonal entries are ghosts or R . | I G | = R by the nonsingularity of I G . Also, for any matrix A and any quasi-identity, I G , we have AI G = A + A G , where A G = AZ G ∈ M n ( G ).There is another notion to help us out. Definition 4.4.
The ( i, j ) - minor A ′ i,j of a matrix A = ( a i,j ) is obtained by deleting the i row and j columnof A . The adjoint matrix adj( A ) of A is defined as the transpose of the matrix ( a ′ i,j ) , where a ′ i,j = (cid:12)(cid:12) A ′ i,j (cid:12)(cid:12) . Remark 4.5. (i)
Suppose A = ( a i,j ) . An easy calculation using Formula (4.1) yields | A | = n X j =1 a i,j a ′ i,j , ∀ i. (4.2) Consequently, a i,j a ′ i,j ≤ ν | A | for each i, j . (ii) If we take k = i, then replacing the i row by the k row in A yields a matrix with two identical rows;thus, its determinant is a ghost, and we thereby obtain n X j =1 a i,j a ′ k,j ∈ G , ∀ k = i ; (4.3) Likewise n X j =1 a j,i a ′ j,k ∈ G , ∀ k = i. One easily checks that adj( B ) adj( A ) = adj( AB ) for any 2 × A and B . However, this failsfor larger n , cf. [IzR3, Example 4.7]. We do have the following fact, which illustrates the subtleties of thesupertropical structure, cf. [IzR3, Proposition 5.6]: Proposition 4.6. adj( AB ) = adj( B ) adj( A ) + ghost . Definition 4.7.
For | A | invertible, define I A = A adj( A ) | A | , I ′ A = adj( A ) | A | A. The matrices I A and I ′ A are quasi-identities, as seen in [IzR3, Theorem 4.13]. The main technique of proofis to define a string (from the matrix A ) to be a product a i ,j · · · a i k ,j k of entries from A and, given such astring, to define its digraph to be the graph whose edges are ( i , j ) , . . . , ( i k , j k ) , counting multiplicities. A k - multicycle in a digraph is the union of disjoint simple cycles, the sum of whose lengths is k ; thus everyvertex in an n -multicycle appears exactly once. A careful examination of the digraph in conjunction withHall’s Marriage Theorem yields the following major results from [IzR3, Theorem 4.9 and Theorem 4.12]: Theorem 4.8. (i) | A adj( A ) | = | A | n . (ii) | adj( A ) | = | A | n − . In case A is a nonsingular, we define A ∇ = adj( A ) | A | . Thus AA ∇ = I A , and A ∇ A = I ′ A . Note that I ′ A and I A may differ off the diagonal, although I A A = AA ∇ A = AI ′ A . This result is refined in [IzR4, Theorem 2.18]. One might hope that A adj( A ) A = | A | A, but this is falsein general! The difficulty is that one might not be able to extract an n -multicycle from a i,j a ′ k,j a k,ℓ . (4.4)For example, when n = 3, the term a , ( a , a , ) a , = a , a ′ , a , does not contain an n -multicycle. We dohave the following positive result from [IzR4, Theorem 4.18]: Theorem 4.9. adj( A ) adj( adj( A )) adj( A ) ∼ = ν | A | n − adj( A ) for any n × n matrix A . The supertropical Hamilton-Cayley theorem.Definition 4.10.
Define the characteristic polynomial f A of the matrix A to be f A = | λI + A | , and the tangible characteristic polynomial to be a tangible polynomial c f A = λ n + P ni =1 b α i λ n − i , where ˆ α i are tangible and b α i ∼ = ν α i , such that f A = λ n + P ni =1 α i λ n − i . Under this notation, we see that α k ∈ R arises from the dominant k -multicycles in the digraph of A . Wesay that a matrix A satisfies a polynomial f ∈ R [ λ ] if f ( A ) ∈ M n ( G ) . Theorem 4.11. ( Supertropical Hamilton-Cayley , [IzR3, Theorem 5.2] ) Any matrix A satisfies both itscharacteristic polynomial f A and its tangible characteristic polynomial c f A . Tropical dependence.
Now we apply supertropical matrix theory to vectors. As in classical math-ematics, one defines a module (often called semi-module in the literature) analogously to module inclassical algebra, noting again that negation does not appear in the definition. It is convenient to stipulatethat the module V has a zero element V , and then we need the axiom: a V = V for all a ∈ R. Also, if ∈ R then we require that v = V for all v ∈ V .In what follows, F always denotes a 1-semifield. In this case, a module over F is called a (supertropical) vector space . The natural example is F ( n ) , with componentwise operations. As in the classical theory, thereis the usual familiar correspondence between the semiring M n ( F ) and the linear transformations of F ( n ) .For v = ( v , . . . , v n ) , w = ( w , . . . , w n ) ∈ F ( n ) , we write v | gs = w when v i | gs = w i for all 1 ≤ i ≤ n. Here is an application of the adjoint matrix, used to solve equations.
Remark 4.12.
Suppose A is nonsingular, and v ∈ F ( n ) . Then the equation Aw = v + ghost has the solution w = A ∇ v. Indeed, writing I A = I + Z G for a ghost matrix Z G , we have Aw = AA ∇ v = I A v = ( I + Z G ) v | gs = v. This leads to the supertropical analog of Cramer’s rule [IzR4, Theorem 3.5]:
LGEBRAIC STRUCTURES 11
Theorem 4.13. If A is a nonsingular matrix and v is a tangible vector, then the equation Ax | gs = v has asolution over F which is the tangible vector having value A ∇ v. Our next task is to characterize singularity of a matrix A in terms of “tropical dependence” of its rows.In some ways the standard supertropical theory works well with matrices, since we are interested mainly inwhether or not this matrix is nonsingular, i.e., if its determinant is tangible; at the outset, at least, we arenot concerned with the precise ghost layer of the determinant. Definition 4.14.
A subset W ⊂ F ( n ) is tropically dependent if there is a finite sum P α i w i ∈ G ( n ) , witheach α i ∈ T , but not all of them R ; otherwise W ⊂ F ( n ) is called tropically independent . A vector v ∈ F ( n ) is tropically dependent on W if W ∪ { v } is tropically dependent. By [IzKR2, Proposition 4.5], we have:
Proposition 4.15.
Any n + 1 vectors in F ( n ) are tropically dependent. Theorem 4.16. ( [IzR3, Theorem 6.5] ) Vectors v , . . . , v n ∈ F ( n ) are tropically dependent, iff the matrixwhose rows are v , . . . , v n is singular. Corollary 4.17.
The matrix A ∈ M n ( F ) over a supertropical domain F is nonsingular iff the rows of A are tropically independent, iff the columns of A are tropically independent.Proof. Apply the theorem to | A | and | A t | , which are the same. (cid:3) There are two competing supertropical notions of base of a vector space, that of a maximal independentset of vectors, and that of a minimal spanning set, but this is unavoidable since, unlike the classical theory,these two definitions need not coincide.4.5.
Tropical bases and rank.
The customary definition of tropical base, which we call s-base (for spanning base ), is a minimal spanning set (when it exists). However, this definition is rather restrictive,and a competing notion provides a richer theory.
Definition 4.18. A d-base (for dependence base ) of a vector space V is a maximal set of tropicallyindependent elements of V . A d,s-base is a d-base which is also an s-base. The rank of a set B ⊆ V ,denoted rank( B ) , is the maximal number of d -independent vectors of B . Our d-base corresponds to the “basis” in [MS, Definition 5.2.4]. In view of Proposition 4.15, all d-basesof F ( n ) have precisely n elements.This leads us to the following definition. Definition 4.19.
The rank of a vector space V is defined as: rank( V ) := max (cid:8) rank( B ) : B is a d-base of V (cid:9) . We have just seen that rank( F ( n ) ) = n. Thus, if V ⊂ F ( n ) , then rank( V ) ≤ n .We might have liked rank( V ) to be independent of the choice of d-base of V , for any vector space V . This isproved in the classical theory of vector spaces by showing that dependence is transitive. However, transitivityof dependence fails in the supertropical theory, and, in fact, different d-bases may contain different numbersof elements, even when tangible. An example is given in [MS, Example 5.4.20], and reproduced in [IzKR2,Example 4.9] as being a subspace of F (4) having d-bases both of ranks 2 and 3. Example 4.20.
The matrix A = has rank 2, but is “ghost annihilated” by the tropicallyindependent vectors v = (1 , , t and v = (1 , , t ; i.e., Av = Av = (5 ν , ν , ν ) t , although > . We do have some consolations.
Proposition 4.21 ([IzKR2, Proposition 4.11]) . For any tropical subspace V of F ( n ) and any tangible v ∈ V, there is a tangible d-base of V containing v whose rank is that of V . Proposition 4.22 ([IzKR2, Proposition 4.13]) . Any n × n matrix of rank m has ghost annihilator of rank ≥ n − m . Semi-additivity of rank.
Definition 4.23.
A function rank S : S → N is monotone if for all S ⊆ S ⊆ S we have rank S ( S ∪ { s } ) − rank S ( S ) ≥ rank S ( S ∪ { s } ) − rank S ( S ) (4.5) for all s ∈ S . Note that (4.5) says that rank S ( S ) − rank S ( S ) ≥ rank S ( S ∪ { s } ) − rank S ( S ∪ { s } ) . Also, taking S = ∅ yields rank S ( S ∪ { s } ) − rank S ( S ) ≤ Lemma 4.24. If rank S : S → N is monotone, then rank S ( S ) + rank S ( S ) ≥ rank S ( S ∪ S ) + rank S ( S ∩ S ) (4.6) for all S , S ⊂ S. Proof.
Induction on m = rank S ( S \ S ). If m = 0 , i.e., S ⊆ S , then the left side of (4.6) equals the rightside. Thus we may assume that m ≥ . Pick s in a d-base of S \ S . Let S ′ = S \ { s } . Noting thatrank S ( S ′ \ S ) = m − , we see by induction thatrank S ( S ) + rank S ( S ′ ) ≥ rank S ( S ∪ S ′ ) + rank S ( S ∩ S ′ ) , (4.7)or (taking S ∪ S ′ instead of S in (4.5)),rank S ( S ) − rank S ( S ∩ S ) = rank S ( S ) − rank S ( S ∩ S ′ ) ≥ rank S ( S ∪ S ′ ) − rank S ( S ′ ) ≥ rank S ( S ∪ S ) − rank S ( S ) , yielding (4.6). (cid:3) Proposition 4.25. rank( S ) + rank( S ) ≥ rank( S ∪ S ) + rank( S ∩ S ) for all S , S ⊂ S. Proof. rank is a monotone function, since each side of (4.5) is 0 or 1, depending on whether or not s isindependent of S i , and only decreases as we enlarge the set. (cid:3) Supertropical eigenvectors.
The standard definition of an eigenvector of a matrix A is a vector v ,with eigenvalue β , satisfying Av = βv . It is well known [BrR] that any (tangible) matrix has an eigenvector. Example 4.26.
The characteristic polynomial f A of A = (cid:18) (cid:19) is ( λ + 4)( λ + 1) + 0 = ( λ + 4)( λ + 1) , and the vector (4 , is a eigenvector of A , with eigenvalue 4. However,there is no eigenvector having eigenvalue 1. In general, the lesser roots of the characteristic polynomial are “lost” as eigenvalues. We rectify thisdeficiency by weakening the standard definition.
Definition 4.27.
A tangible vector v is a generalized supertropical eigenvector of a (not necessarilytangible) matrix A , with generalized supertropical eigenvalue β ∈ T , if A m v | gs = β m v for some m ; theminimal such m is called the multiplicity of the eigenvalue (and also of the eigenvector). A supertropicaleigenvector is a generalized supertropical eigenvector of multiplicity 1. Example 4.28.
The matrix A = (cid:18) (cid:19) of Example 4.26 also has the tangible supertropical eigenvector v = (0 , , corresponding to the supertropical eigenvalue , since Av = (4 ν ,
5) = 1 v + (4 ν , −∞ ) . Proposition 4.29. If v is a tangible supertropical eigenvector of A with supertropical eigenvalue β , thematrix A + βI is singular (and thus β must be a (tropical) root of the characteristic polynomial f A of A ). Conversely, we have:
Theorem 4.30 ([IzR3, Theorem 7.10]) . Assume that ν | T : T → G is 1:1. For any matrix A , the dominanttangible root of the characteristic polynomial of A is an eigenvalue of A , and has a tangible eigenvector. Theother tangible roots are precisely the supertropical eigenvalues of A . LGEBRAIC STRUCTURES 13
Let us return to our example A = (cid:18) (cid:19) . Its characteristic polynomial is λ + 2 λ + 2 = ( λ + 0)( λ + 2) , whose roots are 2 and 0. The eigenvalue 2 has tangible eigenvector v = (0 ,
2) since Av = (2 ,
4) = 2 v , butthere are no other tangible eigenvalues. A does have the tangible supertropical eigenvalue 0, with tangiblesupertropical eigenvector w = (2 , , since Aw = (2 , ν ) = 0 w + ( −∞ , ν ) . Note that A + 0 I = (cid:18) ν
01 2 (cid:19) issingular, because | A + 0 I | = 2 ν .Furthermore, A = (cid:18) (cid:19) is a root of λ + 4 A, and thus A is a root of g = λ + 4 λ = ( λ ( λ + 2)) , but 0is not a root of g although it is a root of f A . This shows that the naive formulation of Frobenius’ theoremfails in the supertropical theory, and is explained in the work of Adi Niv [N].4.7. Bilinear forms and orthogonality.
One can refine the study of bases by introducing angles, i.e.,orthogonality, in terms of bilinear forms. Let us quote some results from [IzKR2].
Definition 4.31. A (supertropical) bilinear form B on a (supertropical) vector space V is a function B : V × V → F satisfying B ( v + v , w + w ) | gs = B ( v , w ) + B ( v , w ) + B ( v , w ) + B ( v , w ) ,B ( αv , w ) = αB ( v , w ) = B ( v , αw ) , for all α ∈ F and v i ∈ V, and w j ∈ V ′ . We work with a fixed bilinear form B = h , i on a (supertropical) vector space V ⊆ F ( n ) . The Grammatrix of vectors v , . . . , v k ∈ F ( n ) is defined as the k × k matrix e G ( v , . . . , v k ) = h v , v i h v , v i · · · h v , v k ih v , v i h v , v i · · · h v , v k i ... ... . . . ... h v k , v i h v k , v i · · · h v k , v k i . (4.8)The set { v , . . . , v k } is nonsingular (with respect to B ) when its Gram matrix is nonsingular.In particular, given a vector space V with s-base { b , . . . , b k } , we have the matrix e G = e G ( b , . . . , b k ),which can be written as ( g i,j ) where g i,j = h b i , b j i . The singularity of e G does not depend on the choice ofs-base. Definition 4.32.
For vectors v, w in V , we write v ⊥⊥ w when h v, w i ∈ G , that is h v, w i | gs = F , and saythat v is left ghost orthogonal to w . We write W ⊥⊥ for { v ∈ V : v ⊥⊥ w for all w ∈ W. } Definition 4.33.
A subspace W of V is called nondegenerate (with respect to B ), if W ⊥⊥ ∩ W is ghost.The bilinear form B is nondegenerate if the space V is nondegenerate. Lemma 4.34.
Suppose { w , . . . , w m } tropically spans a subspace W of V , and v ∈ V. If P mi =1 β i h v, w i i ∈ G for all β i ∈ T , then v ∈ W ⊥⊥ . Theorem 4.35. ( [IzKR2, Theorem 6.7] ) Assume that vectors w , . . . , w k ∈ V span a nondegenerate sub-space W of V . If | e G ( w , . . . , w k ) | ∈ G , then w , . . . , w k are tropically dependent. Corollary 4.36.
If the bilinear form B is nondegenerate on a vector space V , then the Gram matrix (withrespect to any given supertropical d,s-base of V ) is nonsingular. Definition 4.37.
The bilinear form B is supertropically alternate if h v, v i ∈ G for all v ∈ V. B is supertropically symmetric if h v, w i + h w, v i ∈ G for all v, w ∈ V . We aim for the supertropical version ([IzKR2, Theorem 6.19]) of a classical theorem of Artin, that anybilinear form in which ghost-orthogonality is symmetric must be a supertropically symmetric bilinear form.
Definition 4.38.
The (supertropical) bilinear form B is orthogonal-symmetric if it satisfies the followingproperty for any finite sum, with v i , w ∈ V : X i h v i , w i ∈ G iff X i h w, v i i ∈ G , (4.9) B is supertropically orthogonal-symmetric if B is orthogonal-symmetric and satisfies the additionalproperty that h v, w i ∼ = ν h w, v i for all v, w ∈ V satisfying h v, w i ∈ T . The symmetry condition extends to sums, and after some easy lemmas we obtain ([IzKR2, Theorem 6.19]):
Theorem 4.39.
Every orthogonal-symmetric bilinear form B on a vector space V is supertropically sym-metric. Identities of semirings, especially matrices
The word “identity” has several interpretations, according to its context. First of all, there are well-known matrix identities such as the Hamilton-Cayley identity which says that any matrix is a root of itscharacteristic polynomial.Since the classical theory of polynomial identities is tied in with invariant theory, we also introduce layeredpolynomial identities (PIs), to enrich our knowledge of layered matrices.5.1.
Polynomial identities of semirings † . We draw on basic concepts of polynomial identities, i.e., PI’s,say from [R2, Chapter 23]. Since semirings † do not involve negatives, we modify the definition a bit. Definition 5.1.
The free N -semiring † N { x , x , . . . } is the monoid semiring † of the free (word) monoid { x , x , . . . } over the commutative semiring † N . Definition 5.2. A (semiring † ) polynomial identity (PI) of a semiring † R is a pair ( f, g ) of (noncom-mutative) polynomials f ( x , . . . , x m ) , g ( x , . . . , x m ) ∈ N { x , . . . , x m } for which f ( r , . . . , r m ) = g ( r , . . . , r m ) , ∀ r , . . . , r m ∈ R. We write ( f, g ) ∈ id( R ) when ( f, g ) is a PI of R . Remark 5.3. A semigroup identity of a semigroup S is a pair ( f, g ) of (noncommutative) monomials f ( x , . . . , x m ) , g ( x , . . . , x m ) ∈ N { x , . . . , x m } for which f ( s , . . . , s m ) = g ( s , . . . , s m ) , ∀ s , . . . , s m ∈ S . If S is contained in the multiplicative semigroup of a semiring † R , the semigroup identities of S are precisely thesemiring † PIs ( f, g ) where f and g are monomials. Akian, Gaubert and Guterman [AkGG, Theorem 4.21] proved their strong transfer principle , whichimmediately implies the following easy but important observation:
Theorem 5.4. If f, g ∈ N { x , . . . , x n } have disjoint supports and f − g is a PI of M n ( Z ) , then f = g isalso a semiring † PI of M n ( R ) for any commutative semiring † R .Proof. Since Z is an infinite integral domain, f − g is also a PI of M n ( C ) , where C = Z [ ξ , ξ , . . . ]denotes the free commutative ring in countably many indeterminates, implying ( f, g ) is a semiring † PIof M n ( N [ ξ , ξ , . . . ]). But the semiring † M n ( R ) is a homomorphic image of M n ( N [ ξ , ξ , . . . ]), implying( f, g ) ∈ id( M n ( R )). (cid:3) Corollary 5.5.
Any PI of M n ( Z ) yields a corresponding semiring † PI of M n ( R ) for all commutativesemirings † R .Proof. Take f to be the sum of the terms having positive coefficient, and g to be the sum of the terms havingnegative coefficient, and apply the theorem. (cid:3) Many (but not all) matrix PIs can be viewed in terms of Theorem 5.4, although semiring versions of basicresults such as the Amitsur-Levitzki Theorem and Newton’s Formulas often are more transparent here.We say that polynomials f ( x , . . . , x m ) and g ( x , . . . , x m ) are a t - alternating pair if f and g are inter-changed whenever we interchange a pair x i and x j for some 1 ≤ i < j ≤ t. For example, x x and x x are a2-alternating pair. Sometimes we write the non-alternating variables as y , y , . . . ; we write y as shorthandfor all the y j . LGEBRAIC STRUCTURES 15
Definition 5.6.
We partition the symmetric group S t of permutations in t letters into the even permu-tations S + t and the odd permutations S − t . Given a t -linear polynomial h ( x , . . . , x t ; y ) , we define the t -alternating pair h +alt ( x , . . . , x t ; y ) := X σ ∈ S + t h ( x σ (1) , . . . , x σ ( t ) ; y ) and h − alt ( x , . . . , x t ; y ) := X σ ∈ S − t h ( x σ (1) , . . . , x σ ( t ) ; y ) . The standard pair is Stn t := ( h +alt , h − alt ) , where h = x · · · x t . Explicitly,
Stn t := (cid:18) X σ ∈ S + t x σ (1) · · · x σ ( t ) , X σ ∈ S − t x σ (1) · · · x σ ( t ) (cid:19) . The
Capelli pair is Cap t := ( h +alt , h − alt ) , where h = x y x y · · · x t y t . Explicitly,
Cap t := (cid:18) X σ ∈ S + t x σ (1) y x σ (2) y · · · y t − x σ ( t ) y t , X σ ∈ S − t x σ (1) y x σ (2) y · · · y t − x σ ( t ) y t (cid:19) . Proposition 5.7.
Any t -alternating pair ( f, g ) is a PI for every semiring † R spanned by fewer than t elements over its center.Proof. Suppose R is spanned by { b , b , . . . , b t − } . We need to verify f (cid:18) X α i, b i , . . . , X α i,t b i t , . . . (cid:19) = g (cid:18) X α i, b i , . . . , X α i,t b i t , . . . (cid:19) . Since f and g are linear in these entries, it suffices to verify f ( b i , . . . , b i t , . . . ) = g ( b i , . . . , b i t , . . . ) (5.1)for all i , . . . , i t . But by hypothesis, two of these must be equal, say i k and i k ′ , so switching these twoyields (5.1) by the alternating hypothesis. (cid:3) Let e i,j denote the matrix units. The semiring † version of the Amitsur-Levitzki theorem [AmL], thatStn n ∈ id( M n ( N )), is an immediate consequence of Theorem 5.4, and its minimality follows from: Lemma 5.8.
Any pair of multilinear polynomials f ( x , . . . , x m ) and g ( x , . . . , x m ) having no common mono-mials do not comprise a PI of M n ( R ) unless m ≥ n .Proof. Rewriting indices we may assume that x · · · x m appears as a monomial of f, but not of g, and wenote (for ℓ = (cid:2) m (cid:3) + 1) that f ( e , , e , , e , , e , , . . . , e k − ,k , e k,k , . . . ) = e ,ℓ = 0 , but g ( e , , e , , e , , e , , . . . , e k − ,k , e k,k , . . . ) = 0 . (cid:3) Likewise, the identical proof of [R2, Remark 23.14] shows that the Capelli pair Cap n is not a PI of M n ( C ) , and in fact ( e , , ∈ Cap n ( M n ( R )) for any semiring † R. Surpassing identities.
The surpassing identity f | gs = g holds when f ( a , . . . , a m ) | gs = g ( a , . . . , a m )for all a , . . . , a m ∈ R . Example 5.9.
Take the general × matrix A = (cid:18) a bc d (cid:19) . Then tr( A ) = a + d and | A | = ad + bc.A = (cid:18) a + bc b ( a + d ) c ( a + d ) bc + d (cid:19) , so A + adI = (cid:18) a ( a + d ) + bc b ( a + d ) c ( a + d ) bc + d ( a + d ) (cid:19) = tr( A ) A + bcI, implying A + | A | I = tr( A ) A + bc ν I, yielding the surpassing identity A + | A | I | gs = tr( A ) A for × matrices. We might hope for a surpassing identity involving alternating terms in the Hamilton-Cayley polynomial,but a cursory examination of matrix cycles dashes our hopes.
Example 5.10.
Let A = − d ac − −− b − . Then A = cd ab −− cd acbc − − and A = abc cd acdc d abc −− bcd abc , implying A = αA + | A | in this case, where α denotes the other coefficient in f A . But for A = a − −− b −− − c we have A + αA + 2 − − −− abc −− − abc = tr( A ) A + | A | , so neither A + αA nor tr( A ) A + | A | necessarily surpasses the other. Layered surpassing identities.
Since we want to deal with general layers, we write 2 a (instead of a ν )for a + a , but note that s (2 a ) = 2 s ( a ) . When working with the layered structure, we can extend the notionof PI from Definition 5.2 by making use of the following relations that arise naturally in the theory.
Definition 5.11.
The L -surpassing relation | L = is given by a | L = b iff either a = b + c with c s ( b ) -ghost ,a = b,a ∼ = ν b with a s ( b ) -ghost . (5.2)It follows that if a | L = b , then a + b is s ( b )-ghost. When a = b , this means a ≥ ν b and a is s ( b )-ghost. Definition 5.12.
The surpassing ( L, ν ) -relation | L ≡ ν is given by a | L ≡ ν b iff a | L = b and a ∼ = ν b. (5.3) The surpassing L -identity f | L = g holds for f, g ∈ Fun( R ( n ) , R ) if f ( a , . . . , a n ) | L = g ( a , . . . , a n ) forall a , . . . , a n ∈ R .The surpassing ( L, ν ) -identity f | L ≡ ν g holds for f, g ∈ Fun( R ( n ) , R ) if f ( a , . . . , a n ) | L ≡ ν g ( a , . . . , a n ) for all a , . . . , a n ∈ R . Layered surpassing identities of commutative layered semirings.
Just as the Boolean algebra satisfiesthe PI x = x, we have some surpassing identities for commutative layered domains † . Proposition 5.13. (Frobenius identity) ( x + x ) m | L ≡ ν x m + x m . Proof.
This is just a restatement of [IzKR5, Remark 5.2]. (cid:3)
Proposition 5.14. ( x + x + x )( x x + x x + x x ) | L ≡ ν ( x + x )( x + x )( x + x ) . More generally, let g = P i x i , g = P i This is just a restatement of [IzR1, Theorem 8.51]. (cid:3) LGEBRAIC STRUCTURES 17 Layered surpassing identities of matrices. We applied the strong transfer principle of Akian, Gaubert, and Guterman [AkGG, Theorem 4.21] to the(standard) supertropical matrix semiring in [IzR4]. We would like to make a similar argument in the layeredcase, but must avoid the following kind of counterexamples, pointed out by Adi Niv: Example 5.15. Suppose A = (cid:18) [1] [2] [2] [10] (cid:19) . Then A = (cid:18) [1] [2] [2] [4] (cid:19) , so | A | = [10] whereas | A | = [8] , which does not N -surpass | A | (and does not even N -surpass | A | ). The difficulty in the example was that some ν -small entry of A has a high layer which provides | A | a highlayer but does not affect the powers of A . There is a version of surpassing which is useful in this context. Definition 5.16. An element c ∈ R is a strong ℓ -ghost (for ℓ ∈ L + ) if s ( c ) ≥ ℓ .The strong ℓ -surpassing relation a | Sℓ = b holds in an L -layered domain † R , if either a = b + c with c a strong ℓ -ghostor a = b. (5.5) We often take ℓ = s ( b ) . In this case b + b | Sℓ = b (as well as b + b | ℓ = b ).The strong ℓ -surpassing relation ( a i,j ) | Sℓ = ( b i,j ) holds for matrices ( a i,j ) and ( b i,j ) , if a i,j | Sℓ = b i,j foreach i, j . We say that a matrix A is ℓ - layered if each entry has layer ≥ ℓ . We are ready for our other two versionsof layered identities. Definition 5.17. The strong ( ℓ, d ) -surpassing identity f | Sℓ ; d = g holds for f, g ∈ Fun( M n ( R ) ( m ) , M n ( R )) if f ( A , . . . , A m ) | S ˜ ℓ = g ( A , . . . , A m ) with ˜ ℓ = ℓ d , for all ℓ -layered matrices A , . . . , A m ∈ M n ( R ) . In the standard supertropical theory we take ℓ = ˜ ℓ = 1 , but in the general layered theory we may needto consider other ℓ . Formally set P ( x , . . . , x ℓ ) = P + − P − and Q ( x , . . . , x ℓ ) = Q + − Q − . We say Q is admissible if the monomials of Q + and Q − are distinct, for each pair ( i, j ).We then obtain the following metatheorem, along the lines of [AkGG] (just as in [IzR4, Theorem 2.4]): Theorem 5.18. Suppose P = Q is a homogeneous matrix identity of M n ( Z ) of degree d , with Q admissible.Then the matrix semiring † M n ( R ) satisfies the strong ( ℓ, d ) -surpassing identity P + + P − | Sℓ ; d = Q + + Q − . Here are some applications. Corollary 5.19. | AB | | Sℓ ; d = | A | | B | for L -layered n × n matrices A and B , where d = 2 n . Given an L -layered matrix A and the polynomial f A := | λI + A | = α n λ n + · · · + α λ + α , we define the polynomial f f A to be c f A = b α n λ n − + · · · + b α λ + b α , where s ( b α i ) = ℓ n − i and b α i ∼ = ν α i . Theorem 5.20. f f A ( A ) | Sℓ ; d = adj( A ) , where d = n − , for any ℓ -layered matrix A .Proof. This is an identity for M n ( Z ) , using the usual determinant. (cid:3) Proposition 5.21. adj( adj( A )) | Sℓ ; d = | A | n − A , where d = n − , for any ℓ -layered matrix A . Questions for further thought: Q1. What are all the semiring † PIs of M n ( R )?Specifically, we have the Specht-like question:Q2. Are all semiring † PIs of M n ( R ) a consequence of a given finite set? Example 5.22. It is shown in [IzM] that the semiring of × matrices over the max-plus algebra satisfiesthe semigroup identity AB A AB AB A = AB A BA AB A. (5.6) The way of proving this identity is essentially based on showing that pairs of polynomials corresponding tocompatible entries in the right and the left product above define the same function. This identification isperformed by using the machinery of Newton polytopes, and thus is valid also for supertropical polynomials.From the results of [IzM] , we also conclude that this identity is minimal. References [AkBG] M. Akian, R. Bapat, and S. Gaubert. Max-plus algebra, In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R.(eds.) Handbook of Linear Algebra . Chapman and Hall, London, 2006.[AkGG] M. Akian, S. Gaubert, and A. Guterman. Linear independence over tropical semirings and beyond. In Tropical andIdempotent Mathematics , G.L. Litvinov and S.N. Sergeev, (eds.), Contemp. Math. J. f¨ur die Reine und angevandteMathematik , 374:168–195, 1984.[Bo] N. Bourbaki, Alg. Comm. VI , § 3, No.1.[BrR] R. A. Brualdi and H. J. Ryser. Combinatorial matrix theory . Cambridge University Press, 1991.[CHWW] G. Cortinas, C. Haesemeyer, M. Walker, and C. Weibel, Toric varieties, monoid schemes, and descent. Preprint, 2010.[DeS] M. Devlin and B. Sturmfels, Tropical convexity . Documenta Mathematica 9 (2004), 1–27, Erratum 205–6.[Gat] A. Gathmann, Tropical algebraic geometry. Jahresbericht der DMV The theory of semirings with applications in mathematics and theoretical computer science , Vol. 54,Longman Sci & Tech., 1992.[ItMS] I. Itenberg, G. Mikhalkin and E. Shustin, Tropical Algebraic Geometry , Oberwolfach Seminars, 35, Birkh¨auser Verlag,Basel, 2007.[Iz] Z. Izhakian. Tropical arithmetic and matrix algebra. Commun. in Algebra Pacific J. of Math. , to appear. (Preprint atarXiv:1008.0025.)[IzKR3] Z. Izhakian, M. Knebusch, and L. Rowen, Dual spaces and bilinear forms in supertropical linear algebra, Linear andMult. Algebra , to appear. (Preprint at arXiv:1201.6481.)[IzKR4] Z. Izhakian, M. Knebusch, and L. Rowen. Layered tropical mathematics, preprint at arXiv:0912.1398. (SubmittedMay 2012)[IzKR5] Z. Izhakian, M. Knebusch, and L. Rowen. Categorical layered mathematics. Contemporary Mathematics , proceedingsof the CIEM Workshop on Tropical Geometry, to appear. (Preprint at arXiv:1207.3487.)[IzM] Z. Izhakian and S. W. Margolis. Semigroup identities in the monoid of 2-by-2 tropical matrices. Semigroup Furom Adv. in Math Commun. in Algebra , 37(11), 3912–3927, 2009.[IzR3] Z. Izhakian and L. Rowen, Supertropical matrix algebra, Israel J. Math. , 182(1), 383–424, 2011.[IzR4] Z. Izhakian and L. Rowen, Supertropical matrix algebra II: Solving tropical equations. Israel J. Math. J. ofAlgebra , 341(1), 125–149, 2011.[IzR6] Z. Izhakian and L. Rowen, Supertropical polynomials and resultants, J. Algebra Basic Algebra , Freeman, 1980.[Ko] V.M. Kopytov, Lattice ordered groups (Russian), Nauka, Moscow (1984)[L1] G. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction. J. of Math.Sciences , 140(3),426–444, 2007.[L2] G. Litvinov, Dequantization of mathematical structures and tropical idempotent mathematics: An introductorylecture. (See next reference), 2012[L3] G. Litvinov and V.P. Maslov, Tropical and idempotent mathematics: International Workshop. Pncelet Laboratoryand Moscow Center for Continuous Mathematical Education, Independent University of Moscow, 2012.[MS] D. Maclagan and B. Sturmfels, Tropical Geometry , preprint, 2009.[M] D. Marker, Model theory: An introduction , Springer Graduate texts in mathematics; 217, 2002.[N] A. Niv, Characteric polynomials of supertropical matrices, Commun. in Algebra , to appear, 2012.[Par] B. Parker. Exploded fibrations, preprint at arXiv: 0705.2408v1, 2007.[Pay1] S. Payne. Fibers of tropicalizations, Arch. Math., 2010 Correction: preprint at arXiv: ?? [math.AG], 2012. LGEBRAIC STRUCTURES 19 [Pay2] S. Payne. Analytification is the limit of all tropicalizations, preprint at arXiv: 0806.1916v3 [math.AG], 2009.[R1] L.H. Rowen. Graduate Algebra: Commutative View . Pure and Applied Mathematics 73, Amer. Math. Soc., 2006.[R2] L.H. Rowen. Graduate algebra: A noncommutative view Amer. Math. Soc., 2008.[Sa] G.E. Sacks, Saturated Model Theory , Mathematical Lecture Noets 80 Benjamin, 1972.[ShSh] E. Sheiner and S. Shnider, An exploded-layered version of Payne’s generalization of Kapranov’s theorem, preprint,2012.[SS] D. E. Speyer and B. Sturmfels, Tropical mathematics, Math. Mag. , 82 (2009), 1631/2-173.[Vi] O. Viro, Hyperfields for tropical geometry I. Hyperfields and dequantization Preprint at arXiv:math.AG/1006.3034v2[W] C. Weibel, EGA for monoids. Preprint, 2010.[WW] H.J. Weinert and R. Wiegandt, On the structure of semifields and lattice-ordered groups, Periodica MathematicaHungaria 32 (1-2) (1996), 129–147. Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel E-mail address : [email protected] Department of Mathematics, NWF-I Mathematik, Universit¨at Regensburg 93040 Regensburg, Germany E-mail address : [email protected] Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel E-mail address ::