tropical arithmetic and matrix algebra
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Tropical Arithmetic and Matrix AlgebraZur Izhakian a ba Department of Mathematics , Bar-Ilan University , Ramat-Gan, Israelb CNRS et Université Denis Diderot (Paris 7) , Paris, FrancePublished online: 28 Mar 2009.
To cite this article: Zur Izhakian (2009) Tropical Arithmetic and Matrix Algebra, Communications in Algebra, 37:4, 1445-1468,DOI: 10.1080/00927870802466967
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Communications in Algebra®, 37: 1445–1468, 2009Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802466967
TROPICAL ARITHMETIC AND MATRIX ALGEBRA
Zur IzhakianDepartment of Mathematics, Bar-Ilan University, Ramat-Gan,Israel and CNRS et Université Denis Diderot (Paris 7), Paris, France
This article introduces a new structure of commutative semiring, generalizing thetropical semiring, and having an arithmetic that modifies the standard tropicaloperations, i.e., summation and maximum. Although our framework is combinatorial,notions of regularity and invertibility arise naturally for matrices over this semiring;we show that a tropical matrix is invertible if and only if it is regular.
Key Words: Commutative semiring; Max-Plus algebra; Tropical algebra.
2000 Mathematics Subject Classification: Primary 15A09, 15A15, 16Y60; Secondary 15A33, 20M18,51M20.
INTRODUCTION
Traditionally, researchers have been able to frame mathematical theories usingformal structures provided by algebra; geometry is often a source for interestingphenomena in the core of these theories. The semiring structure introduced inthis article emerges from the combinatorics within max-plus algebra and itscorresponding polyhedral geometry, called tropical geometry. Although our groundstructure is a semiring, much of the theory of standard commutative algebra canbe formulated on this semiring, leading to application in combinatorics, semigrouptheory, polynomials algebra, and algebraic geometry.
Tropical mathematics takes place over the tropical semiring �R ∪�−���max�+�, the real numbers equipped with the operations of maximum andsummation, respectively, addition and multiplication (Gathmann, 2006; Itenberg,2007; Richter-Gebert et al., 2005), and it interacts with a number of fields ofstudy including algebraic geometry, polyhedral geometry, commutative algebra,and combinatorics. Polyhedral complexes, resembling algebraic varieties over afield with real non-archimedean valuation, are the main objects of the tropicalgeometry, where their geometric combinatorial structure is a maximal degenerationof a complex structure on a manifold.
Over the past few years, much effort has been invested in the attempt tocharacterize a tropical analogous to classical linear algebra, (Develin et al., 2004;
Received July 31, 2007; Revised March 31, 2008. Communicated by M. Cohen.Address correspondence to Zur Izhakian, Department of Mathematics, Bar-Ilan University,
Ramat-Gan 52900, Israel; Fax: +972(0)9-7442038. E-mail: [email protected]
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Kim and Roush, 2005; Richter-Gebert et al., 2005), and to determine connectionsbetween the classical and the tropical worlds (Mikhalkin, 2006; Shustin, 2005;Speyer and Sturmfels, 2004a). Despite the progress that has been achieved in thesetropical studies, some fundamental issues have not been settled yet; the idempotencyof addition in �R ∪ �−���max�+� is maybe one of the main reasons for that.Addressing this reason, and other algebro-geometric needs, our goals are:
(a) Introducing a new structure of a partial idempotent semiring having its ownarithmetic that generalizes the max-plus arithmetic and also carries a tropicalgeometric meaning;
(b) Presenting a novel approach for a theory of matrix algebra over partialidempotent semirings that includes notions of regularity and semigroupinvertibility, analogous as possible to that of matrices over fields.
The latter goal is central issue in the study of Green’s relations over semigroupsand is essential toward developing a linear representations of semigroups. Our newapproach answers these goals and paves a way to treat other needs like havingnotions of linear dependence and rank.
Our new structure, which we call extended tropical semiring, is built on thedisjoint union of two copies of R, denoted R and R�, together with the formalelement −� that serves as the gluing point of R and R�. Thus,
T �= R ∪ �−�� ∪ R�
is provided with an order, ≺, extending the usual order on R, and endowed withthe addition ⊕ and the multiplication � that modify the familiar operations maxand +. By this setting, �T�⊕��� has the structure of a commutative semiring, ⊕ isidempotent only on R�, and �T�⊕��� allows to define a homomorphic relation toa field with real non-archimedean valuation. From the point of view of algebraicgeometry, ⊕ encodes an additive multiplicity that enables to define tropical algebraicsets in a natural manner.
The second part of the article focuses mainly on introducing a theory of matrixalgebra over �T�⊕���, reassembling the classical theory of matrices over fields, thatincludes notions of regularity and invertibility in a natural way with the followingrelation.
Theorem 3.7. A tropical matrix is pseudo invertible if and only if it is tropicallyregular.
We provide also an explicit characterization of the pseudo-inverse matrix ��
of a regular matrix �, which turns out to be similar to that of the classical theory.Concerning semigroup theory, we show that the monoid Mn�T� of matrices over�T�⊕��� can be related to as an E-dense monoid in which our invertibility suitsE-denseness, that is, the products ��� and ��� are idempotent matrices (Margolisand Pin, 1987).
1. EXTENDED TROPICAL ARITHMETIC — A NEW APPROACH
With two goals in minds, geometrically and algebraically derived, our objectiveis to introduce a new concept of idempotent semiring extensions, applied here to
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the classical tropical semiring �R ∪ �−���max�+�, including also the relation tonon-Archimedean fields with real valuations. Although related topics have beendiscussed earlier for �R ∪ �−���max�+�, cf. Butkovic (2003), Cuninghame-Greenand Butkovic (2004), Develin et al. (2004), and Speyer and Sturmfels (2004b), in thisarticle we use a different approach implemented on a semiring structure having amodified arithmetic. We open by describing the standard tropical framework, thenwe present the basics of our new concept and the associated semiring structure.
1.1. The Tropical Semiring
Tropical mathematics is the mathematics over idempotent semirings, thetropical semiring is usually taken to be �R ∪ �−���max�+�; the real numberstogether with the formal element −�, and with the operations of tropical additionand tropical multiplication
a+ b �= max�a� b�� a · b �= a+ b�
cf. Mikhalkin (2006) and Richter-Gebert et al. (2005). We write R for R ∪ �−��and equip R
�= R with the Euclidean topology, assuming that R is homeomorphic
to �0���. The tropical semiring contains the max-plus algebra (Cuninghame-Greenand Butkovic, 2004; Richter-Gebert et al., 2005) and it emerges as a target of non-Archimedean fields with real valuation; it is an idempotent semiring, i.e., a+ a = a,with the unit �R �= 0, and the zero element OR �= −�.
Elements of the semiring R�1� � n� are called tropical polynomials in nvariables over R and are of the form
f = maxi∈�
�� � i� + �i� � (1.1)
where �·� ·� stands for the standard scalar product, � ⊂ Zn is a finite nonempty setof points i = �i1� � in� with non-negative coordinates, �i ∈ R for all i ∈ �, and =�1� � n�. The addition and multiplication of polynomials are defined accordingto the familiar law.
Any tropical polynomial f ∈ R�1� � n�\�−�� determines a piecewise linearconvex function f̃ � R�n� −→ R. But, in the tropical case, the map f �→ f̃ is notinjective, and one can reduce the polynomial semiring so as to have only thoseelements needed to describe functions.
A tropical hypersurface is defined to be the domain of non-differentiability,also called the corner locus, of f̃ for some f ∈ R�1� � n�\�−��. Therefore,points of a tropical hypersurface can be specified as the points on which the valueof f̃ is attained by at least two monomials of f . This property is crucial forunderstanding the purpose of incorporating additive multiplicities, it will be usedlater to distinguish the corner locus from the other points of a domain.
One of our goals is to establish a semiring structure that allows one to realize(algebraically) the points of a corner locus as a “zero” locus of a polynomial;namely, to have the ability to form algebraic sets. Therefore, we would like to havea structure that not only provides the operation of maximum, but also encodes anindication about its additive multiplicity. In other words, in some sense, to “resolve”the idempotency of �R�max�+�.
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Remark 1.1. Indeed, to address this goal, one may suggest an alternativearithmetic that defines the addition of two equal elements to be −�, which wewrite as “a+ a” = −�, and the addition of different elements to be their maximum.We denote this structure as �R� “max ”�+�. Unfortunately, this type of additionis not associative; for example, for b < a we have “b + �a+ a�” = “b + �−��” = bwhile “�b + a�+ a” = “�a�+ a” = −�.
Our next development addresses this algebro-geometric issue; later we showthat it also servers a solid base for developing a theory of matrix algebra oversemirings that have the notions of regularity and invertibility.
1.2. The Extended Tropical Semiring
Roughly speaking, the central idea of our new approach is a generalizationof �R�max�+� to a semiring structure having a partial idempotent addition thatdistinguishes between sums of similar elements and sums of different elements. Settheoretically, our semiring is composed from the disjoint union of two copies of R,denoted R and R�, which glued along the formal element −� to create the set
T �= R ∪ �−�� ∪ R�
In what follows, we denote the unions R ∪ �−�� and R� ∪ �−�� respectively by Rand R
�, write T× for T\�−��, and call the elements of R reals.
We use the generic notation that a� b ∈ R for reals, a�� b� ∈ R� where a� b ∈ R,and x� y ∈ T. Thus, T is provided with the following order ≺ extending the usualorder on R.
Axiom 1.2. The order ≺ on T is defined as:
(a) −� ≺ x� ∀x ∈ T×;(b) For any real numbers a < b, we have a ≺ b� a ≺ b�, a� ≺ b, and a� ≺ b�;(c) a ≺ a� for all a ∈ R.
One can verify that the corresponding partial order, , holds only in the caseswhere both elements are in R or both are in R�.
Example 1.3. Assume a < b < c are reals; then
−� ≺ a ≺ a� ≺ b ≺ b� ≺ c ≺ c�
According to the rules of ≺, cf. Axiom 1.2, T is then endowed with the twooperations ⊕ and �, addition and multiplication respectively, defined as below.(We use the notation max≺ to denote the maximum with respect to the order ≺.)
Axiom 1.4. The laws of the extended tropical arithmetic are:
(1) −�⊕ x = x⊕−� = x for each x ∈ T;(2) x⊕ y = max≺�x� y� unless x = y;
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(3) a⊕ a = a� ⊕ a� = a�;(4) −�� x = x�−� = −� for each x ∈ T;(5) a� b = a+ b for all a� b ∈ R;(6) a� � b = a� b� = a� � b� = �a+ b��.
We call the triple �T�⊕��� the extended tropical semiring; later we show that�T�⊕��� indeed have the structure of commutative semiring with unit �T �= 0 andOT �= −�.
Recall that two preliminary essential demands have been required on ⊕,validity of associativity and, simultaneously, differentiation between addition ofsimilar reals and addition of different reals. The first requirement is satisfied byAxiom 1.2 (3) and Axiom 1.4 (2); that is, for reals, we have
b ⊕ a� = b ⊕ �a⊕ a�?=↑�b ⊕ a�⊕ a =
�a�⊕ a = a�� a � b�
�a��⊕ a = a�� a = b�
�b�⊕ a = b� b � a�
(1.2)
the equality is then derived from Axiom 1.4 (2).
Remark 1.5.
(1) The addition ⊕ (in comparison to that of �R�max�+�) is not idempotent, sincea⊕ a = a�; this is one of the main aspects of our approach.
(2) R�is an ideal of T, where sometimes we want to think about as a set of pseudo-
zeros, namely, consisting of those elements to be ignored. On the other hand, byAxiom 1.4 (3), R
�can be also realized as a “shadow” copy of R whose elements
carry additive multiplicities >1, received as tropical sums of identical reals. Thisview is important for understanding the linkage between our arithmetic and thenotion of tropicalization.
In the context of semigroups, both �T�⊕� and �T��� are monoids but notgroups and thus, invertibility is invalid for both ⊕ and �. Yet, for �, one can talkabout partial invertibility which is well defined on reals only.
Definition 1.6. The division, denoted ��, of x� y ∈ T, with y �= −�, is defined asx�� y = x� �−y�, where −y = �−a�� when y = a�.
Note that �� is not well defined over all T, but suits our purpose. The cancellationlaw, x� y = x� z⇒ y = z, does not always hold; for example, the equality a� �b = a� � b� does not satisfy cancellation.
Remark 1.7. The structure of �T�⊕��� has been formulated on two disjoint copiesof R with the modification of the operations max and +; the same construction canbe performed for any idempotent semiring with the property a+ b ∈ �a� b� and inparticular for �Z�max�+�.
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1.3. Properties of the Extended Tropical Arithmetic
Having formulated the extended tropical arithmetic, we address its basicproperties. We describe only the main cases in detail; therefore, the trivial casesinvolving −� are omitted. To clarify the exposition, sometimes, we treat theelements of R and R� separately.
Commutativity: Axiomatic (cf. Axiom 1.4).
Associativity: By definition �a⊕ b�� = a� ⊕ b� and �a� b�� = a� � b�. Thus,for different elements in T, the associativity of ⊕ and � is clear by the associativityof max and + which also provides the associativity of � for all T. The case in whichidentical reals are involved has already been examined in (1.2). For the case of twosimilar elements in R� we have
�a⊕ b��⊕ c� ={b� ⊕ c� = �b ⊕ c��� b � a�
a⊕ c� ↘ a � b�
={c�� c � a� b�
a� a � b� c�
and
a⊕ �b� ⊕ c�� = a⊕ �b ⊕ c�� =
c�� c � a� b�
b�� b � a� c�
a� a � b� c�
which have equal evaluations. (The other cases of compound expressions areobtained by the same way.)
Distributivity: To verify distributivity of � over ⊕, for the case when allelements are reals, write
a� �b ⊕ c� =
a� b� b � c�
a� b�� b = c�
a� c� c � b�
and
�a� b�⊕ �a� c� =
a� b� b � c�
�a� b��� b = c�
a� c� c � b�
and compare the evaluations with respect to the different ordering of the involvedarguments. When elements of both R and R� are involved, use the abovespecification together with Axiom 1.4; for example,
a� � �b ⊕ c� = �a� �b ⊕ c���
= ��a� b�⊕ �a� c���
= �a� b�� ⊕ �a� c�� = �a� b��⊕ �a� c��
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TROPICAL ARITHMETIC AND MATRIX ALGEBRA 1451
Zero: By definition OT �= −� is the additive identity of T (cf. Axiom 1.4 (1)),and it annihilates T (cf. Axiom 1.4 (4)).
One: One can easily check that �T �= 0 is the multiplicative identity of T.
Theorem 1.8. The set T equipped with the addition ⊕ and the multiplication � is a(nonidempotent) commutative semiring, �R��⊕� is an additive semigroup, and �R���and �R���� are multiplicative semigroups.
Remark 1.9. In view of Axiom 1.4, � is realized as the onto order preservingprojection
� � �T�⊕��� −→ (R��⊕��)� (1.3)
where � � a �→ a�, � � a� �→ a�, and � � −� �→ −�. Then, � is a semiringhomomorphism, and we write x� for the image of x ∈ T in R
�, where � is is the
identity for each x ∈ R�. Accordingly, call a� the �-value of a. Given x� y ∈ T, we
say that x is greater than y, or maximal, up to � if x� � y�, similarly, when x� = y�,we say that x and y are equal up to �.
Writing xn for the tropical product x� x� · · · � x of n factors, we have thefollowing lemma.
Lemma 1.10. �x⊕ y�n = xn ⊕ yn, n ∈ N, for any x� y ∈ T.
Proof. Assume n > 1; by induction,
�x⊕ y�n = �x⊕ y��x⊕ y�n−1 = �x⊕ y��xn−1 ⊕ yn−1� = xn ⊕ xn−1y ⊕ xyn−1 ⊕ yn
Suppose x � y; then
xn � xn−1y ⊕ xyn−1 ⊕ yn � yn�
and �x⊕ y�n = xn. Similarly, if y � x, then �x⊕ y�n = yn. In the case of x = y, wehave x⊕ y ∈ R
�, xn ⊕ yn ∈ R
�, and xn ⊕ yn = xn = �x⊕ y�n. �
Corollary 1.11. �⊕s
i=1 xi�n =⊕s
i=1 xni � n ∈ N� for any x1� � xs ∈ T.
Corollary 1.12. The “Cauchy” inequality
x1 � x2 � · · · � xn xn1 ⊕ xn2 ⊕ · · · ⊕ xnn
holds for any x1� � xn ∈ T; equality occurs only if ��x1� = ��x2� = · · · = ��xn� and atleast one xi is in R
�.
1.4. Tropical Arithmetics and Tropicalization
The informal term tropicalization is used to describe a map, based on a realvaluation, of objects defined over a non-Archimedean field K with real valuation
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to objects defined over �R�max�+�; objects are either varieties or polynomials.The tropicalization of a variety W ⊂ K�n� is a polyhedral complex in R�n�, while apolynomial in n variables in K�1� � n� is mapped to a tropical polynomial inn variables in R�1� � n�, which we recall determines an affine piecewise linearfunction.
Let K be an algebraically closed field with a real non-Archimedean valuation
Val � �K�+� ·� −→ �R�max�+�� (1.4)
for example, assume K is the field of locally convergent complex Puiseux series, ofthe form
f�t� =∑a∈R
cata� ca ∈ C�
where R ⊂ Q is bounded from below and the elements of R have a boundeddenominator. Then,
Val�f� ={−min�a ∈ R � ca �= 0�� f ∈ K�1� � n�\0�−�� f = 0�
(1.5)
is a real valuation satisfying the rules of being non-Archimedean,
�i� Val�f · g� = Val�f�+ Val�g��(1.6)
�ii� Val�f + g� ≤ max�Val�f��Val�g��
(Note that Val is not a homomorphism, since it does not preserve associativity.)Thus, in the sense of tropicalization, the arithmetic operations of K are replacedwith the correspondence: · �→ + and + �→ max .
Remark 1.13. Taking f� g ∈ K with Val�f� = Val�g� = a, then Val�f + g� can beany point of the ray �−�� a�. These cases provide the motivation for the use of�T�⊕��� as the target of Val that allows to distinguish between the cases in whichFormula (1.6)(ii) is interpreted as equality and the cases it is inequality.
In order to realize �T�⊕��� as the target of Val, to each point a� ∈ R�, weassign the ray Pa� �= �−�� a� and to each a ∈ R we assign the singleton Pa �= �a�,in particular P−� �= �−��; therefore x ∈ Px for each x ∈ T. With this construction,we obtain the inclusions:
P−�� Pa ⊂ Pa�� Pa� ⊂ Pb� ⇐⇒ a ≺ b� ∀a� b ∈ R (1.7)
(Recall that two series in K that are vanished in order 1 must vanish on order atleast 1; the inclusions (1.7) address this property.)
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Let G�T� �= �Px � x ∈ T�, then Val�f� ∈ Px for some Px ∈ G�T� which clearlyneeds not be unique. Accordingly, for each pair f ∈ K and x ∈ T, we define therelation
Val�f� ∈ Px or Val�f� � Px (1.8)
determined by the inclusion of Val�f� in Px.
Theorem 1.14. Formula (1.8) yields a homomorphism; that is, for any f� g ∈ K withVal�f� ∈ Px and Val�g� ∈ Py, we have Val�f · g� ∈ Px�y and Val�f + g� ∈ Px⊕y.
Proof. Suppose Val�f� = a, Val�g� = b. Then, since x ∈ Px for each x ∈ T,
Val�f · g� = Val�f�+ Val�g� = a� b ∈ Pa�b�
and Val�f · 0� = Val�f�+ Val�0� = a� �−�� ∈ P−�. For the additive relation,write
Val�f + g� ≤ max�Val�f�� Val�g��
= max�a� b� =
a ∈ Pa = Pa⊕b a > b�
a ∈ Pa ⊂ Pa� = Pa⊕a a = b�
b ∈ Pb = Pa⊕b b > a�
and use the inclusion Pa ⊂ Pa� , cf. (1.7). The case of Val�f + 0� is trivial. �
1.5. The Relation to the Max-Plus Arithmetic
The structure of �T�⊕��� provides a much richer structure, generalizing boththe max-plus semiring and the one suggested in Remark 1.1, and achieves the bestof both worlds.
Lemma 1.15. The map
� � �T�⊕��� −→ �R�max�+�� (1.9)
� � a� �→ a, � � a �→ a, and � � −� �→ −�, is a semiring epimorphism.
Proof. Clearly, � is onto. Assume ��x� = a and ��y� = b, where x� y ∈ T, then��x⊕ y� = max�a� b� = max���x�� ��y�� and ��x� y� = a+ b = ��x�+ ��y�. �
On the other hand one can also define the following lemma.
Lemma 1.16. The map
� � �R�max�+� −→ �R��⊕���� (1.10)
� � a �→ a� and � � −� �→ −�, is a semiring isomorphism that embeds �R�max�+� in�T�⊕���.
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Proof. Take a� b ∈ R, then ��max�a� b�� = �max�a� b��� = a� ⊕ b� = ��a�⊕ ��b��and ��a+ b� = �a+ b�� = a� � b� = ��a�� ��b�. R
� ⊂ T, so � embeds �R�max�+�in �T�⊕���. �
Corollary 1.17. Categorically, by Remark 1.9, Lemmas 1.15, and 1.16, the diagram
commutes.
Corollary 1.17 displays �T�⊕��� as a generalization of �R�max�+� which isendowed with a richer structure in the sense that it encodes an indication about theadditive multiplicity of elements in R. Namely, since a⊕ a = a� and a⊕ a� = a�,a� can be realized as a point with additive multiplicity >1. (Clearly, computationsfor �R�max�+� can be performed on �T�⊕��� and then to be sent back to�R�max�+�.)
As for the arithmetic suggested in Remark 1.1 (i.e., defined with “a+ a” =−�), one may suggest the map
� � �T�⊕��� −→ �R� “max ”�+�� (1.11)
� � a �→ a, � � a� �→ −�, and � � −� �→ −�; but, since �R� “max ”�+� is notassociative, � is not a homomorphism.
1.6. Geometric View
Let us remind that one of our goals was to obtain a semiring structure thatenables us to treat algebraically the points of a corner locus of tropical functions,namely, to define tropical algebraic set. To present only the frame of this idea, givena tropical polynomial f ∈ T�1� � n�, we define the tropical algebraic set of thecorresponding function f̃ to be
Z�f̃ � = {�x1� � xn� ∈ T�n� � f̃ �x1� � xn� ∈ R
�}
Therefore, the corner locus of f ∈ R�1� � n� over �R�max�+� is just therestriction of Z�f̃ �, considered as a polynomial over �T�⊕���, to the real points,i.e., Z�f̃ � ∩ R
�n�.
Example 1.18. Consider the similar linear functions f�x� = x⊕ a over �T�⊕���and f�x� = max�x� a� over �R�max�+�, see Fig. 1. Restricting the domain to R only,over �T�⊕���, the image of the corner locus, which contains the single point a, isdistinguished and is now mapped to R�.
The study of polynomial algebras and tropical algebraic sets over �T�⊕���will be treated in a forthcoming article.
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Figure 1 The graph of the linear function f̃ �x� = x + a over �T�⊕���, on the left-hand side, and thecorresponding function f̃ �x� = max�x� a� over �R�max�+�, on the right-hand side.
2. MATRIX ALGEBRA
Our forthcoming study is dedicated to introducing the fundamentals of thematrix algebra over �T�⊕��� whose operations of are typically combinatorial. Yet,developing an algebraic theory, analogous to classical theory of matrix algebra overfields, with a view to combinatorics, is our main goal. This goal is supported by theconnections to graph theory (Lawler, 1976), the theory of automata (Margolis andPin, 1987), and semiring theory (Golan, 1992).
Notations. For the rest of the article, assuming the nuances of the differentarithmetics are already familiar, we write xy for the product x� y, x
yfor the division
x�� y, and xn for x� · · · � x repeated n times.
2.1. Tropical Matrices
It is standard that if R is a semiring, then we have the semiring Mn�R� of n× nmatrices with entries in R, where addition and multiplication are induced from R asin the familiar matrix construction. Accordingly, we define the semiring of tropicalmatrices Mn�T� over �T�⊕���, whose unit is the matrix
� = 0 −�
−� 0
� (2.1)
and whose zero matrix is � = �−��� ; therefore, Mn�T� is also a multiplicativemonied. We write � = �aij� for a tropical matrices � ∈ Mn�T� and denote the entriesof � as aij . Since T is a commutative semiring, x� = �x for any x ∈ T and � ∈Mn�T�.
As in the familiar way, we define the transpose of � = �aij� to be �t = �aji�,and have the relation in the following proposition.
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1456 IZHAKIAN
Proposition 2.1. ����t = �t�t
(The proof is standard by the commutativity and the associativity of ⊕ and �over T.)
The minor �ij is obtained by deleting the i row and j column of �. We definethe tropical determinant to be
��� = ⊕�∈Sn
�a1��1� · · · an��n��� (2.2)
where Sn is the set of all the permutations on �1� � n�. Equivalently, ��� can bewritten in terms of minors as
��� =⊕j
aioj��ioj
�� (2.3)
for some fixed index io. Indeed, in the classical sense, since parity of indices’ sumsare not involved in Formula (2.2), the tropical determinant is a permanent, whichmakes the tropical determinant a pure combinatorial function. The adjoint matrixAdj��� of � = �aij� is defined as the matrix �a′ij�
t, where a′ij = ��ij�.Observation 2.2. The tropical determinant has the following properties:
(1) Transposition and reordering of rows or columns leave the determinantunchanged;
(2) The determinant is linear with respect to scalar multiplication of any given rowor column.
2.2. Regularity of Matrices
Using the special structure of �T�⊕���, the algebraic formulation ofcombinatorial properties becomes possible.
Definition 2.3. A matrix � ∈ Mn�T� is said to be tropically singular, or singular,for short, whenever ��� ∈ R
�, otherwise � is called tropically regular, or regular, for
short.
In particular, when two or more different permutations, � ∈ Sn, achieve the�-value of ��� simultaneously, or the permutation that reaches the �-value of ���involves an entry in R
�, then � is singular.
Remark 2.4. Despite some classical properties hold for the tropical determinant,cf. Observation 2.2, the familiar relation ���� = ������ does not hold true on oursetting; for example, take the matrix
� =(1 12 3
)with �2 = �� =
(3 45 6
)� (2.4)
then, ��� = 4 and ������ = 8, while ��2� = 9�. In the view of tropicalization, whichignores signs, the determinant of a matrix over K is assigned to the permanent of
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a matrix in Mn�T�; this explains the tropical situation in which the product of tworegular matrices might be singular.
Theorem 2.5. A matrix with two identical rows or columns is singular.
Proof. Proof by induction on n ≥ 2. The case of n = 2 is clear by directcomputation. Assume the two first columns of � are identical, and expand ���in terms of minors along the first row, that is ��� =⊕
i a1i��1i�. Since a11 = a12
and �11 = �12, then a11��11� = a12��12�, and so a11��11� ⊕ a12��12� ∈ R�. By the
induction hypothesis, for any i > 2, �1i is a matrix with identical columns, and issingular, that is ��1i� ∈ R
�for all i > 2. When adding all together, a1i��1i� ∈ R
�for
all i = 1� � n, and thus ��� ∈ R�. �
Theorem 2.6. If � and � are regular matrices and their product �� is also regular,then ���� = ������. When either � or � is singular, then �� is also singular.
Proof. Let Sn be the set of all the permutations on N = �1� � n�, and letFn = �N → N� be the set of all maps from N to itself, in particular Sn ⊂ Fn.Denoting the entries of �� by �ab�ij , we write the determinant ���� in the form ofFormula (2.2) as:
���� = ⊕�∈Sn
⊙i
�ab�i��i� =⊕�∈Sn
⊙i
(⊕k
�aikbk��i��
)= ⊕
�∈Sn
(�a11b1��1� ⊕ · · · ⊕ a1nbn��1�� · · · �an1b1��n� ⊕ · · · ⊕ annbn��n��
)= ⊕
�∈Sn
⊕�∈Fn
(⊙i
�ai��i�b��i���i��
)= ⊕
�∈Sn
⊕�∈Fn
(⊙i
ai��i�
⊙i
b��i���i�
) (∗)
By the structure of the left-hand side of �∗�, we can see that the value of ����is obtained when both
⊙i ai��i� and
⊙i b��i���i� attain their maximal evaluation at
the same time. We show that this is possible. Namely, both reach their maximalevaluation on the same �, which we denote by �o; the corresponding � ∈ Sn is thendenoted by �o. Note that when ���� ∈ R there must be exactly one pair, �o and �o;otherwise, by definition, �� would not be regular.
Case I: Suppose �o ∈ Sn is a permutation which maximizes⊙
i ai��i�. Weshow that there is also a permutation �o ∈ Sn that maximizes
⊙i b��i���i� for �o.
Assume ���� ∈ R, with �t ∈ Sn maximizes⊙
j bj�t�j�. Generally speaking, for anygiven � ∈ Sn and �t ∈ Sn, there exits � ∈ Sn which makes the diagram
commutative, where we use the notation �·� to indicate the appropriate indices.Accordingly, choosing �o ∈ Sn for which �o � �o = �t, we obtain
⊙i b�o�i��o�i� =
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1458 IZHAKIAN⊙j bj�t�j�; in this case the two components of �∗� reach their maximum
simultaneously, and we can write
�∗� =(⊙
i
ai�o�i�
)(⊙j
bj�o�j�
)=
( ⊕�∈Sn
⊙i
ai��i�
)(⊕�∈Sn
⊙j
bj��j�
)= ������
When � is singular, there are at least two different �1� �2 ∈ Sn that attain the�-value of ��� or a single � ∈ Sn that involves a nonreal entry. The latter case isobvious, since �∗� has a nonreal multiplier and thus ���� ∈ R
�. Suppose �1� �2 ∈ Sn
are two permutations satisfying �o = �l � �l, l = 1� 2, then
�∗� =⊙i
ai�1�i�
⊙i
b�1�i��1�i� =⊙i
ai�2�i�
⊙i
b�2�i��2�i��
and hence ���� ∈ R�.
Case II: Suppose �o ∈ Fn\Sn, ���� ∈ R, and let �o ∈ Sn be the correspondingpermutation which maximizes the product⊙
i
�ai��i�b��i���i�� =⊙i
ai��i�
⊙i
b��i���i� (∗∗)
in �∗�. In particular, there is only one such pair, �o and �o, for otherwise �� wouldnot be regular. Since �o � Sn, there are at least two indices i1 �= i2 with �o�i1� =�o�i2� = ko. Let h1 �= �o�i1� and h2 �= �o�i2�; then h1 �= h2, since �o ∈ Sn. Subject to�o and �o, �∗∗� can be rewritten as(⊙
i
ai�o�i�
)(⊙i
b�o�i��o�i�
)=
(⊙i
ai�o�i�
)((b�o�i1���o�i1�b�o�i2��o�i2�
) ⊙i �=i1i2
b�o�i��o�i�
)
=(⊙
i
ai��o�i�
)((bkoh1bkoh2
) ⊙i �=i1i2
b�o�i��o�i�
) (∗∗∗)
Denote by �̃o ∈ Sn the permutation obtained by switching between the images ofi1 and i2 in �o, while all other correspondences remain as they are; explicitly,�̃o�i1� = h2, �̃o�i2� = h1, and �̃o�i� = �o�i�, for all i �= i1� i2. With respect to �o, wehave the following combinatorial situation:
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TROPICAL ARITHMETIC AND MATRIX ALGEBRA 1459
(The diagram helps us to understand the modification of �o.) Using �̃, we expand�∗∗∗� further,
�∗∗∗� =(⊙
i
ai�o�i�
)((b�o�i2��̃o�i2�b�o�i1��̃o�i1�
) ⊙i �=i1i2
b�o�i��̃o�i�
)
=(⊙
i
ai�o�i�
)(⊙i
b�o�i��̃o�i�
)�
which means that �̃o �= �o also attain the �-value of ���� and thus ���� ∈ R�. This
contradicts the assumption that �o � Sn, so �o ∈ Sn, and this case has already beendiscussed before. �
Example 2.7. Take the matrices
� =(1 22 3
)and � =
(3 10 2
)� then �� =
(4 45 5
)
� is singular with ��� = 4�, � is regular with ��� = 5, and �� is singular with���� = 9�. On the other hand, �2 = � 6 4
3 4 � is regular with ��2� = 10, so ������ = ��2�.
3. INVERTIBILITY OF MATRICES
We introduce a new notion of semigroup invertibility, and present it for thematrix monoid Mn�T�; this type of invertibility can be adopted to any abstractsemigroup having a distinguished subset. Although our framework is typicallycombinatorial, we show how classical results are carried naturally on our setting.
3.1. Basic Definitions
We open with an abstract definition.
Definition 3.1. Let � be semigroup, and let U ⊂ S be a proper subset with theproperty that for any u ∈ U there exists some v ∈ U for which vu ∈ U and uv ∈ U .We call U a distinguished subset of S.
An element x ∈ � is said to be pseudo-invariable if there is y ∈ � for whichxy ∈ U and yx ∈ U , in particular all the members of U are pseudo-invertible. WhenU consists of all idempotents elements of �, the pseudo-invertibility is then calledE-denseness (Margolis and Pin, 1987). A monoid is called E-dense if all of itselements are E-denseness.
To emphasize, for the purpose of pseudo-invertibility, U needs not be closedunder the law of �. The notion of E-denseness is already known in literature, whilethe weaker version of pseudo-invertibility is new.
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To apply the notion of pseudo-invariability to Mn�T�, viewed as monoid, wedefine a pseudo-unit matrix to be a regular matrix of the form
�̃ = 0 ��ij
��ji 0
(3.1)
that is, �ij ∈ R�for all i �= j, and �ii = 0, for each i = 1� � n. Since �̃ is regular,
we necessarily have ��̃ � = 0, and in particular the unit matrix � , cf. (2.1), is also apseudo unit. We define the distinguished subset Un�T� ⊂ Mn�T� to be
Un�T� ={�̃ � �̃ is a pseudo unit matrix
} (3.2)
Therefore, � ∈ Un�T� and hence ��̃ = �̃� = �̃ , for each �̃ ∈ Un�T�, which makesUn�T� a distinguished subset satisfies the condition of Definition 3.1.
Correspondingly, we define the distinguished subset Uidemn �T� ⊂ Mn�T� to be
U idmn �T� = {
�̃ � �̃ is an idempotent pseudo unit matrix} (3.3)
Remark 3.2. It is easy to show that any �̃ ∈ U2�T� is idempotent. For n > 2, notall of the pseudo-units are idempotents; for example, take the triangular matrix
�̃ = 0 a� b�
−� 0 c�
−� −� 0
�
with a�c� � b�.
Using Un�T�, we explicitly define pseudo invertibility on Mn�T�:
Definition 3.3. A matrix � ∈ Mn�T� is said to be pseudo invertible if there exits amatrix � ∈ Mn�T� such that �� ∈ Un�T� and �� ∈ Un�T�. If � is pseudo invertible,then we call � a pseudo-inverse matrix of � and denote it as ��.
We use the notation of �� since the pseudo matrix needs not be unique;moreover, in our setting ��� is not necessarily equal to ���, and thus might beevaluated for different pseudo-units.
Example 3.4. Consider the following matrices:
� = 0 −2 −1−2 0 �−3��−1 �−3�� 0
and �′ = 0 −2 −1−2 0 −3−1 −3 0
For these matrices we have ��′ ∈ Un�T�, �′� ∈ Un�T�, and also �� ∈ Un�T�.Namely, � has at least two pseudo-inverses.
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Remark 3.5. For the case of M2�T�, our notion of pseudo-invertibility coincideswith the notion of general invertibility in a semigroup in the sense of Von-Neumannregularity (Lallement, 1979), but not for Mn�T� with n > 2.
Remark 3.6. When one intends to use the other semirings structures, either�R�max�+� or �R� “max ”�+�, in order to define an inverse matrix, or pseudo-inverse, it appears to be very restricted or even impossible. Over �R�max�+�,unless −� is involved, the zero element −� is unreachable by tropical sums andproducts of entries. Thus, obtaining the unit matrix � as products of matrices is veryrestricted. On the other hand, when using �R� “max ”�+�, as suggested in Remark1.1, multiplication is not associative, which makes implementation very difficult.
3.2. Theorem on Tropical Pseudo-Inverse Matrix
Theorem 3.7. A matrix � ∈ Mn�T� is pseudo invertible if and only if is tropicallyregular. In case � is regular, �� can be defined as
�� = Adj���
���Before proving the theorem, we recall some definitions and present new notation:
(a) Division in T is denoted by ÷ and interpreted as the substraction a− b in theclassical sense. We write a−1 for 0
aand am for the tropical product of a repeated
m times (which is just m · a in the usual sense);(b) We use the notation �ih�jk for ��ij�hk, that is, �h� k�-minor of the minor �ij ,
where h �= i and k �= j with respect to the initial indices of �. Accordingly, ��ij�is written in terms of minors as ��ij� =
⊕k �=j ahk��ih�jk�, where h �= i
Proof. We prove only multiplication on right, ��� ∈ Un�T�; the multiplication onleft is proved in the same way.
�⇐� Assume ��� ∈ R�, and at the same time, there exists a pseudo-inverse
��. Then, by Theorem 2.6, ����� ∈ R�and their product is singular. Recalling that
��̃ � = 0 for all �̃ ∈ Un�T�, we have ��� � Un�T�.
�⇒� We write �� for the adjoint matrix Adj���, for short, and denote theproduct ��� by � = �bij�. Assuming � is regular, we need to prove that ���
��� ∈Un�T�, or equivalently, that ��� = ����̃ for some �̃ ∈ Un�T�. To prove this, we needto verify the following conditions:
(1) bii = ��� for each i;(2) bij ∈ R
�, for any i �= j;
(3) � ���� � = 0.
Diagonal entries: When i = j,
bii =⊕k
aika�ki =
⊕k
aik��ik� = ���� (3.4)
since this is just the expansion of ��� along row i (cf. Eq. (2.3)).
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Non-diagonal entries: For i �= j,
bij =⊕k
aika�kj =
⊕k
aik��jk� ∈ R�� (3.5)
since this is the expansion of the determinant of the matrix obtained from � byreplacing row j with a copy of row i, and which therefore has two identical rows,and is singular (Theorem 2.5).
Regularity of product: To prove � ���� � = 0, we show equivalently that ��� =
���n. Let Sn be the set of all permutations on N = �1� � n�, and let Fn = �N → N�
be the set of all maps from N to itself, i.e., Sn ⊂ Fn, and write the expansion of ���explicitly,
��� = ⊕�∈Sn
(⊙i
bi��i�
)= ⊕
�∈Sn
(⊙i
(⊕k
aik����i�k�))
= ⊕�∈Sn
(�a11����1�1� ⊕ · · · ⊕ a1n����1�n�� · · · �an1����n�1� ⊕ · · · ⊕ ann����n�n��
)= ⊕
�∈Sn
⊕�∈Fn
⊙i
�ai��i�����i���i��� (3.6)
Assume �o ∈ Sn and �o ∈ Mn achieve the �-value of ���. In case �o is theidentity, by Eq. (3.4), bii = ���, for each i, and thus,⊙
i
bi�o�i� =⊙i
bii = ���n (3.7)
Otherwise, when �o is not the identity, we write
c �=⊙i
�ai�o�i����o�i��o�i�
��� (3.8)
for the product that reaches the �-value of (3.6) and prove it always ≺ ���n.
Case I: Assume �o ∈ Sn is a permutation, then Formula (3.8) can bereordered to the form
a1�o�1���1�o�1�
� · · · an�o�n���n�o�n�
� (∗)
If �∗� � ���n, then it must have at least one component aj�o�n���j�o�n�
� � ���, but thiscontradicts the maximality of ���. On the other hand, if all aj�o�n�
��j�o�n�� = ���, we
get a contradiction to the regularity of ���. Therefore, �∗� ≺ ���n.Case II: Assume �o ∈ Fn\Sn, then there exist at least two indices i1 �= i2 for
which �o�i1� = �o�i2� = jo. We show the existence of a permutation �l ∈ Sn thatreaches the same �-value for Formula 3.8 as �o reaches, the proof is then completedby Case I, applied to �l.
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For the two components ai1jo���o�i1�jo
� and ai2jo���o�i2�jo
� of (3.8), indexed by�o�i1� = �o�i2� = jo, we have the following combinatorial layouts:
The diagrams are useful to understand the modification of �o.Since � ∈ Mn\Sn, there exists at least one index jh �= jo in N\Im��o�. Therefore,
the corresponding component, aijh���o�i�jh
�, is absent in (3.8). Without loss ofgenerality, we take ai2jo
���o�i2�jo� and modify it. Clearly, ���o�i2�jo
� involves an entryaj̇h
, let ih be the index for which ��ih� = jh. Then ���o�i2�jo� = aihjh
���o�i2�ih�jojh�, and
hence, by the maximality of �o,
ai2jo���o�i2�jo
� = ai2joaihjh
���o�i2�ih�jojh� = aihjh
��ihjh�
Namely, we have specified another map �1 ∈ Mn with �1�i1� = jo, �1�i2� = jh, and�1�i� = �o�i� for all i �= i1� i2. Therefore, we reduced the number of indices sharinga same image in �o to have Im��o� ⊂ Im��1� ⊆ N . Proceeding inductively, we geta chain
Im��o� ⊂ Im��1� ⊂ · · · ⊂ Im��l� = N�
the left equality is due to the finiteness of Fn. Thus �l ∈ Sn; the proof of Case II isthen completed by Case I.
So, we have showed that the identity �o is the single permutation thatmaximizes (3.6), and for which we have � = ��� = ����̃ . Since ��� ∈ R and �̃ isregular, so is �. This completes the proof of Theorem 3.7 on pseudo invertibility ofmatrices over �T�⊕���. �
We push the result of Theorem 3.7 further.
Theorem 3.8. For each regular matrix � ∈ Mn�T�, the products ��� and ��� areidempotents.
Proof. Writing �̃ = ���, with �̃ = ��ij�, we prove that �̃ = �̃ 2. Recall that a�ij =
��ji���� , then
�ij =⊕k
aika�kj =
⊕k
aik
��jk���� = aike
��jke�
��� � (3.9)
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1464 IZHAKIAN
for some fixed ke. Suppose ��̃ �2 = ���2�ij �, then
��2�ij =
⊕h
�ih�hj =⊕h
(⊕k
aik
��hk����
)(⊕l
ahl
��jl����
)
=⊕h
⊕k
⊕l
aik
��hk���� ahl
��jl���� � (3.10)
and we need to prove the equality
���⊕k
aik��jk� =⊕h
⊕k
⊕l
aik��hk�︸ ︷︷ ︸�I�
ahl��jl�︸ ︷︷ ︸�II�
(3.11)
To see that ��2�ij � �ij , take h = j to have �II� =⊕l ahl��jl� =
⊕l ajl��jl� = ���.
By way of contradiction, assume ��2�ij � �ij , and suppose ko, ho, and lo are the indicesreaching the �-value of ��2�ij in Formula (3.10), then
aiko��hoko
�aholo��jlo
� � aike��jke
����
Clearly, aike��jke
� � aiko��jko
�, since otherwise we would have a contradiction to themaximality of (3.9). Thus,
aiko��hoko
�aholo��jlo
� � aiko��jko
�����
and hence
��hoko�aholo
��jlo� � ��jko
���� (3.12)
Due to the maximality of ���, we also have ��� � ahoko��hoko
�, and by (3.12) we get
��hoko�aholo
��jlo� � ��jko
���� � ��jko�ahoko
��hoko�
Namely,
aholo��jlo
� � ahoko��jko
� (3.13)
This contradicts the specification of ko as the index that reaches the maximumfor (3.11). This completes the proof that �����2 = ���; the case of multiplicationon left is proved in the same way. �
Corollary 3.9. A matrix � is E-dense in Mn�T�, with respect to U idmn �T�, if and only
if is tropically regular.
Example 3.10. Take the regular matrix
� =(1 −12 2
)� then �� =
(2 −12 1
)�−3��
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TROPICAL ARITHMETIC AND MATRIX ALGEBRA 1465
where ��� = 3. (Recall that, in tropical sense, multiplying by �−3� means dividingby 3.) The product ��� is then(
1 −12 2
)(2 −12 1
)�−3� =
(3 0�
4� 3
)�−3� =
(0 �−3��1� 0
)∈ U idm
n �T�
On the other hand, if we take the singular matrix
� =(1 −14 2
)� then �� =
(2 −14 1
)�−3���
where here ��� = 3�. Computing the product ���, we get(1 −14 2
)(2 −14 1
)�−3�� =
(3� 0�
6� 3�
)�−3�� � Un�T��
which is not a regular matrix, and therefore ��� � Un�T�.
A few immediate conclusions are derived from our last results:
Corollary 3.11. Assume � is a regular matrix, then:
(1) Adj��� is also regular;(2) ��� = ������−1, and if � = �� then ��� = ���� = 0.
Proof. The first assertion is obvious. �, ��, and ��� are all regular, then byTheorem 2.6 ������� = ��̃ � = 0, and hence ��� = ����−1. �
The converse assertion of (2) is not true; for example, take the matrix
� =(−1 −2−2 1
)� then �� =
(1 −2−2 −1
)
Although ��� = ���� = 0, we have � �= ��.
Remark 3.12. Contrary to the classical theory of matrices over fields, tropically,the relation ����� = ���� does not hold true; for example, take the regular matrixas in (2.4), then
� =(1 12 3
)� �� =
(3 12 1
)�−4�� and ���� =
(6 45 3
)�−8�
On the other hand, �2 is not regular, cf. Remark 2.4, and the computation ofAdj��2�/��2� yields
����� =(6 45 3
)�−9��� where �2 =
(3 45 6
)�
this shows that ����2 �= �����.
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1466 IZHAKIAN
3.3. Matrices with Real Entries
Denoting by Mn�R� the semiring of matrices over �R�max�+�, the epimor-phism � � �T�⊕���→ �R�max�+�, cf. (1.9), induces in the standard way theepimorphism
�∗ � Mn�T� −→ Mn�R�
of matrix semirings. We write �∗��� for the image of � ∈ Mn�T� in Mn�R�.Conversely, set-theoretic, Mn�R� ⊂ Mn�T�.
Proposition 3.13. Suppose � ∈ Mn�T� is regular, where both � and �� have only
real entries, ��� = �̃ ′, and ��� = �̃ ′′. Then
�∗��̃′�� = �� �∗��
��̃ ′� = ��� �∗��̃′′��� = ��� and �∗���̃ ′′� = �
Proof. We prove the relation �∗��̃ ′�� = �. Letting �̃ ′ = ��ij�, we show that
�(⊕
k
�ikakj
)= aij�
for all i� j. Recall that �ik = �⊕
h aih��jh�����−1, cf. Formula (3.5), and �ik ∈ R�
whenever i �= k. Composing together, we get
⊕k
(⊕h
aih��jh����−1)akj =
(⊕k�h
aih��jh�akj
)���−1 (∗)
Using Formulas (3.7) and (3.8), we see that the maximal value of ��jh�akj is attainedwhen k = h = j and it is ���. Thus,
���∗�� = ��aij��jj�ajj���−1� = aij
The other relations are proved in the same way. �
Remark 3.14. In the sense of Proposition 3.15, the matrices �̃ ′ and �̃ ′′ are pseudoright/left identities of � and ��, respectively.
Pushing the results of Proposition 3.15 forward, we conclude with thefollowing corollary.
Corollary 3.15. Suppose � ∈ Mn�T� is regular. Let ��� = �̃ ′ and ��� = �̃ ′′; then
�∗��̃ ′�� = �∗���� �∗����̃ ′� = �∗�����
�∗��̃ ′′��� = �∗����� and �∗���̃ ′′� = �∗���
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TROPICAL ARITHMETIC AND MATRIX ALGEBRA 1467
Example 3.16. Let
� =(1 12 3
)� then �� =
(−1 −3−2 −3
)and ��� = �̃ ′ =
(0 �−2��1� 0
)
Computing the products, we have
�̃ ′� =(0 �−2��1� 0
)(1 12 3
)=
(1 1�
2� 3
)�
���̃ ′ =(−1 −3−2 −3
)(0 �−2��1� 0
)=
( −1 �−3���−2�� −3
)�
and it easily verify that �∗��̃ ′�� = � and �∗����̃ ′� = ��.
ACKNOWLEDGMENTS
The author would like to thank Prof. Eugenii Shustin for his invaluable help.I’m deeply grateful to him for his support and the fertile discussions we had.
A part of this work was done during the author’s stay at the Max PlanckInstitut für Mathematik (Bonn). The author is very grateful to Max Planck Institutfor the hospitality and excellent work conditions.
The author has been supported by the Chateaubriand scientific post-doctoratefellowships, Ministry of Science, French Government, 2007–2008.
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