Tropical Arithmetic and Tropical Matrix Algebra
ReCALL (2005)
- arXiv: math/0505458
Available from arxiv.org
or
Abstract
This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, 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.
Author-supplied keywords
Available from arxiv.org
Page 1
Tropical Arithmetic and Tropical Matrix Algebra
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TROPICAL ARITHMETIC AND TROPICAL MATRIX ALGEBRA
ZUR IZHAKIAN
Abstract. This paper introduces a new structure of commutative semiring, generalizing the
tropical semiring, and having an arithmetic that modifies the standard tropical operations, 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.
Introduction
Traditionally, researchers have been able to frame mathematical theories using formal structures
provided by algebra; geometry is often a source for interesting phenomena in the core of these
theories. The semiring structure introduced in this paper emerges from the combinatorics within
max-plus algebra and its corresponding polyhedral geometry, called tropical geometry. Although
our ground structure is a semiring, much of the theory of standard commutative algebra can be
formulated on this semiring, leading to application in combinatorics, semigroup theory, polynomials
algebra, and algebraic geometry.
Tropical mathematics takes place over the tropical semiring (R ∪ {−∞},max,+), the real
numbers equipped with the operations of maximum and summation, respectively, addition and
multiplication [4, 6, 12], and it interacts with a number of fields of study including algebraic
geometry, polyhedral geometry, commutative algebra, and combinatorics. Polyhedral complexes,
resembling algebraic varieties over a field with real non-archimedean valuation, are the main objects
of the tropical geometry, where their geometric combinatorial structure is a maximal degeneration
of a complex structure on a manifold.
Over the past few years, much effort has been invested in the attempt to characterize a tropical
analogous to classical linear algebra, [3, 7, 12], and to determine connections between the classical
and the tropical worlds [11, 13, 14]. Despite the progress that has been achieved in these tropical
studies, some fundamental issues have not been settled yet; the idempotency of 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 own arithmetic
that generalizes the max-plus arithmetic and also carries a tropical geometric meaning;
(b) Presenting a novel approach for a theory of matrix algebra over partial idempotent semir-
ings that includes notions of regularity and semigroup invertibility, analogous as possible
to that of matrices over fields.
The latter goal is central issue in the study of Green’s relations over semigroups and is essential
toward developing a linear representations of semigroups. Our new approach answers these goals
and paves a way to treat other needs like having a notions of linear dependency and rank.
Our new structure, which we call extended tropical semiring , is built on the disjoint union
of two copies of R, denoted R and Rν , together with the formal element −∞ that serves as the
gluing point of R and Rν . Thus,
T := R ∪ {−∞} ∪ Rν
Date: Febuary 2008.
1991 Mathematics Subject Classification. Primary 15A09, 15A15, 16Y60; Secondary 15A33, 20M18, 51M20.
Key words and phrases. Tropical Algebra, Max-Plus Algebra, Commutative Semiring.
The author has been supported by the Chateaubriand scientific post-doctorate fellowships, Ministry of Science,
French Government, 2007-2008.
1
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at
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TROPICAL ARITHMETIC AND TROPICAL MATRIX ALGEBRA
ZUR IZHAKIAN
Abstract. This paper introduces a new structure of commutative semiring, generalizing the
tropical semiring, and having an arithmetic that modifies the standard tropical operations, 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.
Introduction
Traditionally, researchers have been able to frame mathematical theories using formal structures
provided by algebra; geometry is often a source for interesting phenomena in the core of these
theories. The semiring structure introduced in this paper emerges from the combinatorics within
max-plus algebra and its corresponding polyhedral geometry, called tropical geometry. Although
our ground structure is a semiring, much of the theory of standard commutative algebra can be
formulated on this semiring, leading to application in combinatorics, semigroup theory, polynomials
algebra, and algebraic geometry.
Tropical mathematics takes place over the tropical semiring (R ∪ {−∞},max,+), the real
numbers equipped with the operations of maximum and summation, respectively, addition and
multiplication [4, 6, 12], and it interacts with a number of fields of study including algebraic
geometry, polyhedral geometry, commutative algebra, and combinatorics. Polyhedral complexes,
resembling algebraic varieties over a field with real non-archimedean valuation, are the main objects
of the tropical geometry, where their geometric combinatorial structure is a maximal degeneration
of a complex structure on a manifold.
Over the past few years, much effort has been invested in the attempt to characterize a tropical
analogous to classical linear algebra, [3, 7, 12], and to determine connections between the classical
and the tropical worlds [11, 13, 14]. Despite the progress that has been achieved in these tropical
studies, some fundamental issues have not been settled yet; the idempotency of 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 own arithmetic
that generalizes the max-plus arithmetic and also carries a tropical geometric meaning;
(b) Presenting a novel approach for a theory of matrix algebra over partial idempotent semir-
ings that includes notions of regularity and semigroup invertibility, analogous as possible
to that of matrices over fields.
The latter goal is central issue in the study of Green’s relations over semigroups and is essential
toward developing a linear representations of semigroups. Our new approach answers these goals
and paves a way to treat other needs like having a notions of linear dependency and rank.
Our new structure, which we call extended tropical semiring , is built on the disjoint union
of two copies of R, denoted R and Rν , together with the formal element −∞ that serves as the
gluing point of R and Rν . Thus,
T := R ∪ {−∞} ∪ Rν
Date: Febuary 2008.
1991 Mathematics Subject Classification. Primary 15A09, 15A15, 16Y60; Secondary 15A33, 20M18, 51M20.
Key words and phrases. Tropical Algebra, Max-Plus Algebra, Commutative Semiring.
The author has been supported by the Chateaubriand scientific post-doctorate fellowships, Ministry of Science,
French Government, 2007-2008.
1
Page 2
is provided with an order, ≺, extending the usual order on R, and endowed with the addition ⊕
and the multiplication ⊙ that modify the familiar operations max and +. By this setting, (T,⊕,⊙)
has the structure of a commutative semiring, ⊕ is idempotent only on Rν , and (T,⊕,⊙) allows
to define a homomorphic relation to a field with real non-archimedean valuation. From the point
of view of algebraic geometry, ⊕ encodes an additive multiplicity that enables to define tropical
algebraic sets in a natural manner.
The second part of the paper focuses mainly on introducing a theory of matrix algebra over
(T,⊕,⊙), reassembling the classical theory of matrices over fields, that includes notions of regularity
and invertibility in a natural way with the following relation:
Theorem 3.7: A tropical matrix is pseudo invertible if and only if it is tropically regular.
We provide also an explicit characterization of the pseudo inverse matrix A▽ of a regular matrix
A, 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 suits E-denseness, that is the products AA▽ and A▽A are idempotent
matrices [10].
Acknowledgement : The author would like to thank Prof. Eugenii Shustin for his invaluable
help. I’m deeply grateful 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-Planck-Institut fu¨r Mathematik
(Bonn). The author is very grateful to MPI for the hospitality and excellent work conditions.
1. Extended Tropical Arithmetic – A New Approach
With two goals in minds, geometrically and algebraically derived, our objective is to introduce
a new concept of idempotent semiring extensions, applied here to the classical tropical semiring
(R ∪ {−∞},max,+), including also the relation to non-Archimedean fields with real valuations.
Although related topics have been discussed earlier for (R ∪ {−∞},max,+), cf. [1, 2, 3, 15], in
this paper we use a different approach implemented on a semiring structure having a modified
arithmetic. We open by describing the standard tropical framework, then we present the basics of
our new concept and the associated semiring structure.
1.1. The tropical semiring. Tropical mathematics is the mathematics over idempotent semir-
ings, the tropical semiring is usually taken to be (R∪{−∞},max,+ ); the real numbers together
with the formal element −∞, and with the operations of tropical addition and tropical multipli-
cation
a+ b := max{a, b} , a · b := a+ b ,
cf. [11, 12]. 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 [2, 12]
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 1
R¯
:= 0, and the zero element 0
R¯
:= −∞.
Elements of the semiring R¯[λ1, . . . , λn] are called tropical polynomials in n variables over R¯ and
are of the form
(1.1) f = max
i∈Ω
{〈Λ, i〉+ αi} ,
where 〈 ·, · 〉 stands for the standard scalar product, Ω ⊂ Zn is a finite nonempty set of points
i = (i1, . . . , in) with nonnegative coordinates, αi ∈ R for all i ∈ Ω, and Λ = (λ1, . . . , λn). The
addition and multiplication of polynomials are defined according to the familiar law.
Any tropical polynomial f ∈ R¯[λ1, . . . , λn]\{−∞} determines a piecewise linear convex function
f˜ : R(n) −→ R. But, in the tropical case, the map f 7→ f˜ is not injective, and one can reduce the
polynomial semiring so as to have only those elements needed to describe functions.
A tropical hypersurface is defined to be the domain of non-differentiability, also called the corner
locus, of f˜ for some f ∈ R¯[λ1, . . . , λn]\{−∞}. Therefore, points of a tropical hypersurface can be
specified as the points on which the value of f˜ is attained by at least two monomials of f . This
property is crucial for understanding the purpose of incorporating additive multiplicities, it will be
used later to distinguish the corner locus from the other points of a domain.
2
and the multiplication ⊙ that modify the familiar operations max and +. By this setting, (T,⊕,⊙)
has the structure of a commutative semiring, ⊕ is idempotent only on Rν , and (T,⊕,⊙) allows
to define a homomorphic relation to a field with real non-archimedean valuation. From the point
of view of algebraic geometry, ⊕ encodes an additive multiplicity that enables to define tropical
algebraic sets in a natural manner.
The second part of the paper focuses mainly on introducing a theory of matrix algebra over
(T,⊕,⊙), reassembling the classical theory of matrices over fields, that includes notions of regularity
and invertibility in a natural way with the following relation:
Theorem 3.7: A tropical matrix is pseudo invertible if and only if it is tropically regular.
We provide also an explicit characterization of the pseudo inverse matrix A▽ of a regular matrix
A, 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 suits E-denseness, that is the products AA▽ and A▽A are idempotent
matrices [10].
Acknowledgement : The author would like to thank Prof. Eugenii Shustin for his invaluable
help. I’m deeply grateful 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-Planck-Institut fu¨r Mathematik
(Bonn). The author is very grateful to MPI for the hospitality and excellent work conditions.
1. Extended Tropical Arithmetic – A New Approach
With two goals in minds, geometrically and algebraically derived, our objective is to introduce
a new concept of idempotent semiring extensions, applied here to the classical tropical semiring
(R ∪ {−∞},max,+), including also the relation to non-Archimedean fields with real valuations.
Although related topics have been discussed earlier for (R ∪ {−∞},max,+), cf. [1, 2, 3, 15], in
this paper we use a different approach implemented on a semiring structure having a modified
arithmetic. We open by describing the standard tropical framework, then we present the basics of
our new concept and the associated semiring structure.
1.1. The tropical semiring. Tropical mathematics is the mathematics over idempotent semir-
ings, the tropical semiring is usually taken to be (R∪{−∞},max,+ ); the real numbers together
with the formal element −∞, and with the operations of tropical addition and tropical multipli-
cation
a+ b := max{a, b} , a · b := a+ b ,
cf. [11, 12]. 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 [2, 12]
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 1
R¯
:= 0, and the zero element 0
R¯
:= −∞.
Elements of the semiring R¯[λ1, . . . , λn] are called tropical polynomials in n variables over R¯ and
are of the form
(1.1) f = max
i∈Ω
{〈Λ, i〉+ αi} ,
where 〈 ·, · 〉 stands for the standard scalar product, Ω ⊂ Zn is a finite nonempty set of points
i = (i1, . . . , in) with nonnegative coordinates, αi ∈ R for all i ∈ Ω, and Λ = (λ1, . . . , λn). The
addition and multiplication of polynomials are defined according to the familiar law.
Any tropical polynomial f ∈ R¯[λ1, . . . , λn]\{−∞} determines a piecewise linear convex function
f˜ : R(n) −→ R. But, in the tropical case, the map f 7→ f˜ is not injective, and one can reduce the
polynomial semiring so as to have only those elements needed to describe functions.
A tropical hypersurface is defined to be the domain of non-differentiability, also called the corner
locus, of f˜ for some f ∈ R¯[λ1, . . . , λn]\{−∞}. Therefore, points of a tropical hypersurface can be
specified as the points on which the value of f˜ is attained by at least two monomials of f . This
property is crucial for understanding the purpose of incorporating additive multiplicities, it will be
used later to distinguish the corner locus from the other points of a domain.
2
Page 3
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 have a structure that not only provides the operation of
maximum, but also encodes an indication about its additive multiplicity. In other word, in some
sense, to “resolve” the idempotency of (R¯,max,+ ).
Remark 1.1. Indeed, to address this goal, one may suggest an alternative arithmetic that defines
the addition of two equal elements to be −∞, which we write as “a+a” = −∞, and the addition of
different elements to be their maximum. We denote this structure as (R¯, “max ”,+). Unfortunately,
this type of addition is not associative; for example, for b < a we have “b+(a + a) ” = “b+(−∞) ” =
b while “ (b+ a) + a” = “(a) + a” = −∞.
Our next development addresses this algebro-geometric issue; later we show that it also servers
a solid base for developing a theory of matrix algebra over semirings 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 generalization of (R¯,max,+) to a semiring structure having a partial idempotent addition that
distinguishes between sums of similar elements and sums of different elements. Set theoretically,
our semiring is composed from the disjoint union of two copies of R, denoted R and Rν , which
glued along the formal element −∞ to create the set
T := R ∪ {−∞} ∪ Rν .
In what follows we denote the unions R ∪ {−∞} and Rν ∪ {−∞} respectively by R¯ and R¯ν , write
T
× for T \ {−∞}, and call the elements of R reals.
We use the generic notation that a, b ∈ R for reals, aν , bν ∈ Rν where a, b ∈ R, and x, y ∈ T.
Thus, T is provided with the following order ≺ extending the usual order on R:
Axiom 1.2. The order ≺ on T is defined as:
(1) −∞ ≺ x, ∀x ∈ T×;
(2) for any real numbers a < b, we have a ≺ b, a ≺ bν , aν ≺ b, and aν ≺ bν ;
(3) a ≺ aν for all a ∈ R.
One can verify that the corresponding partial order, , holds only in the cases where both elements
are in R or both are in Rν .
Example 1.3. Assume a < b < c are reals; then
−∞ ≺ a ≺ aν ≺ b ≺ bν ≺ c ≺ cν .
According to the rules of ≺, cf. Axiom 1.2, T is then endowed with the two operations ⊕ 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;
(3) a⊕ a = aν ⊕ aν = aν ;
(4) −∞⊙ x = x⊙−∞ = −∞ for each x ∈ T;
(5) a⊙ b = a+ b for all a, b ∈ R;
(6) aν ⊙ b = a⊙ bν = aν ⊙ bν = (a + b)ν .
We call the triple (T,⊕,⊙) the extended tropical semiring ; later we show that (T,⊕,⊙) indeed
have the structure of commutative semiring with unit 1
T
:= 0 and 0
T
:= −∞.
Recall that two preliminary essential demands have been required on ⊕ , validity of associativity
and, simultaneously, differentiation between addition of similar reals and addition of different reals.
3
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 have a structure that not only provides the operation of
maximum, but also encodes an indication about its additive multiplicity. In other word, in some
sense, to “resolve” the idempotency of (R¯,max,+ ).
Remark 1.1. Indeed, to address this goal, one may suggest an alternative arithmetic that defines
the addition of two equal elements to be −∞, which we write as “a+a” = −∞, and the addition of
different elements to be their maximum. We denote this structure as (R¯, “max ”,+). Unfortunately,
this type of addition is not associative; for example, for b < a we have “b+(a + a) ” = “b+(−∞) ” =
b while “ (b+ a) + a” = “(a) + a” = −∞.
Our next development addresses this algebro-geometric issue; later we show that it also servers
a solid base for developing a theory of matrix algebra over semirings 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 generalization of (R¯,max,+) to a semiring structure having a partial idempotent addition that
distinguishes between sums of similar elements and sums of different elements. Set theoretically,
our semiring is composed from the disjoint union of two copies of R, denoted R and Rν , which
glued along the formal element −∞ to create the set
T := R ∪ {−∞} ∪ Rν .
In what follows we denote the unions R ∪ {−∞} and Rν ∪ {−∞} respectively by R¯ and R¯ν , write
T
× for T \ {−∞}, and call the elements of R reals.
We use the generic notation that a, b ∈ R for reals, aν , bν ∈ Rν where a, b ∈ R, and x, y ∈ T.
Thus, T is provided with the following order ≺ extending the usual order on R:
Axiom 1.2. The order ≺ on T is defined as:
(1) −∞ ≺ x, ∀x ∈ T×;
(2) for any real numbers a < b, we have a ≺ b, a ≺ bν , aν ≺ b, and aν ≺ bν ;
(3) a ≺ aν for all a ∈ R.
One can verify that the corresponding partial order, , holds only in the cases where both elements
are in R or both are in Rν .
Example 1.3. Assume a < b < c are reals; then
−∞ ≺ a ≺ aν ≺ b ≺ bν ≺ c ≺ cν .
According to the rules of ≺, cf. Axiom 1.2, T is then endowed with the two operations ⊕ 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;
(3) a⊕ a = aν ⊕ aν = aν ;
(4) −∞⊙ x = x⊙−∞ = −∞ for each x ∈ T;
(5) a⊙ b = a+ b for all a, b ∈ R;
(6) aν ⊙ b = a⊙ bν = aν ⊙ bν = (a + b)ν .
We call the triple (T,⊕,⊙) the extended tropical semiring ; later we show that (T,⊕,⊙) indeed
have the structure of commutative semiring with unit 1
T
:= 0 and 0
T
:= −∞.
Recall that two preliminary essential demands have been required on ⊕ , validity of associativity
and, simultaneously, differentiation between addition of similar reals and addition of different reals.
3
Page 4
The first requirement is satisfied by Axiom 1.2 (3) and Axiom 1.4 (2); that is, for reals, we have
the following:
(1.2) b⊕ aν = b⊕ (a⊕ a) ?=
↑
(b⊕ a)⊕ a =
(a)⊕ a = aν , a ≻ b,
(aν)⊕ a = aν , a = b,
(b)⊕ a = b, b ≻ a,
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, since a⊕a = aν ;
this is one of the main aspects of our approach.
(2) R¯ν 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, by Axiom 1.4 (3), R¯ν can
be also realized as a “shadow” copy of R whose elements carry additive multiplicities > 1,
received as tropical sums of identical reals. This view is important for understanding the
linkage between our arithmetic and the notion of tropicalization.
In the context of semigroups, both (T,⊕) and (T,⊙) are monoids but not groups and thus,
invertibility is invalid for both ⊕ and ⊙ . Yet, for ⊙ , one can talk about partial invertibility which
is well defined on reals only.
Definition 1.6. The division, denoted ⊙▽, of x, y ∈ T, with y 6= −∞, is defined as x ⊙▽ y =
x⊙ (−y), where −y = (−a)ν when y = aν .
Note that ⊙▽ is not well defined over all T, but suits our purpose. The cancellation law,
x⊙ y = x⊙ z ⇒ y = z, does not always hold; for example, the equality aν ⊙ b = aν ⊙ bν does not
satisfy cancellation.
Remark 1.7. The structure of (T,⊕,⊙) has been formulated on two disjoint copies of R with
the modification of the operations max and + ; the same construction can be performed for any
idempotent semiring with the property a+ b ∈ {a, b} and in particular for (Z,max,+ ).
1.3. Properties of the extended tropical arithmetic. Having formulated extended tropical
arithmetic, we address the its basic properties. We describe only the main cases in detail; therefore,
the trivial cases involving −∞ are omitted. To clarify the exposition, sometimes, we treat the
elements of R and Rν separately.
Commutativity: Axiomatic (cf. Axiom 1.4).
Associativity: By definition (a⊕b)ν = aν⊕bν and (a⊙b)ν = aν⊙bν . Thus, for different elements
in T, the associativity of ⊕ and ⊙ is clear by the associativity of max and + which also provides
the associativity of ⊙ for all T. The case in which identical reals are involved has already been
examined in (1.2). For the case of two similar elements in Rν we have:
(a⊕ bν)⊕ cν =
{
bν ⊕ cν = (b⊕ c)ν , b a,
a⊕ cν ց a ≻ b,
=
{
cν , c a, b,
a, a ≻ b, c,
and
a⊕ (bν ⊕ cν) = a⊕ (b ⊕ c)ν =
cν , c a, b,
bν , b a, c,
a, a ≻ b, c,
which have equal evaluations. (The other cases of compound expressions are obtained by the same
way.)
4
the following:
(1.2) b⊕ aν = b⊕ (a⊕ a) ?=
↑
(b⊕ a)⊕ a =
(a)⊕ a = aν , a ≻ b,
(aν)⊕ a = aν , a = b,
(b)⊕ a = b, b ≻ a,
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, since a⊕a = aν ;
this is one of the main aspects of our approach.
(2) R¯ν 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, by Axiom 1.4 (3), R¯ν can
be also realized as a “shadow” copy of R whose elements carry additive multiplicities > 1,
received as tropical sums of identical reals. This view is important for understanding the
linkage between our arithmetic and the notion of tropicalization.
In the context of semigroups, both (T,⊕) and (T,⊙) are monoids but not groups and thus,
invertibility is invalid for both ⊕ and ⊙ . Yet, for ⊙ , one can talk about partial invertibility which
is well defined on reals only.
Definition 1.6. The division, denoted ⊙▽, of x, y ∈ T, with y 6= −∞, is defined as x ⊙▽ y =
x⊙ (−y), where −y = (−a)ν when y = aν .
Note that ⊙▽ is not well defined over all T, but suits our purpose. The cancellation law,
x⊙ y = x⊙ z ⇒ y = z, does not always hold; for example, the equality aν ⊙ b = aν ⊙ bν does not
satisfy cancellation.
Remark 1.7. The structure of (T,⊕,⊙) has been formulated on two disjoint copies of R with
the modification of the operations max and + ; the same construction can be performed for any
idempotent semiring with the property a+ b ∈ {a, b} and in particular for (Z,max,+ ).
1.3. Properties of the extended tropical arithmetic. Having formulated extended tropical
arithmetic, we address the its basic properties. We describe only the main cases in detail; therefore,
the trivial cases involving −∞ are omitted. To clarify the exposition, sometimes, we treat the
elements of R and Rν separately.
Commutativity: Axiomatic (cf. Axiom 1.4).
Associativity: By definition (a⊕b)ν = aν⊕bν and (a⊙b)ν = aν⊙bν . Thus, for different elements
in T, the associativity of ⊕ and ⊙ is clear by the associativity of max and + which also provides
the associativity of ⊙ for all T. The case in which identical reals are involved has already been
examined in (1.2). For the case of two similar elements in Rν we have:
(a⊕ bν)⊕ cν =
{
bν ⊕ cν = (b⊕ c)ν , b a,
a⊕ cν ց a ≻ b,
=
{
cν , c a, b,
a, a ≻ b, c,
and
a⊕ (bν ⊕ cν) = a⊕ (b ⊕ c)ν =
cν , c a, b,
bν , b a, c,
a, a ≻ b, c,
which have equal evaluations. (The other cases of compound expressions are obtained by the same
way.)
4
Page 5
Distributivity : To verify distributivity of ⊙ over ⊕, for the case when all elements are reals, write
a⊙ (b⊕ c) =
a⊙ b, b ≻ c,
a⊙ bν , b = c,
a⊙ c, c ≻ b,
and
(a⊙ b)⊕ (a⊙ c) =
a⊙ b, b ≻ c,
(a⊙ b)ν , b = c,
a⊙ c, c ≻ b,
and compare the evaluations with respect to the different ordering of the involved arguments.
When elements of both R and Rν are involved, use the above specification together with Axiom
1.4; for example,
aν ⊙ (b ⊕ c) = (a⊙ (b⊕ c))ν =
((a⊙ b)⊕ (a⊙ c))ν =
(a⊙ b)ν ⊕ (a⊙ c)ν = (a⊙ bν)⊕ (a⊙ cν) .
Zero: By definition 0
T
:= −∞ 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 1
T
:= 0 is the multiplicative identity of T.
Theorem 1.8. The set T equipped with the addition ⊕ and the multiplication ⊙ is a (non-
idempotent) commutative semiring, (Rν ,⊕) is an additive semigroup, and (R,⊙) and (Rν ,⊙) are
multiplicative semigroups.
Remark 1.9. In the view of Axiom 1.4, ν is realized as the onto order preserving projection
(1.3) ν : (T,⊕,⊙) −→ (R¯ν ,⊕,⊙) ,
where ν : a 7→ aν , ν : aν 7→ aν , and ν : −∞ 7→ −∞. Then, ν is a semiring homomorphism and
we write xν for the image of x ∈ T in R¯ν , where ν is is the identity for each x ∈ R¯ν . Accordingly,
call aν the ν-value of a. Given x, y ∈ T, we say that x is greater than y, or maximal, up to ν if
xν ≻ yν , similarly, when xν = yν we say that x and y are equal up to ν .
Writing xn for the tropical product x⊙ x⊙ · · · ⊙ x of n factors we have:
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, we have x⊕y ∈ R¯ν ,
xn ⊕ yn ∈ R¯ν , and xn ⊕ yn = xn = (x⊕ y)n.
Corollary 1.11. (
⊕s
i=1 xi)
n =
⊕s
i=1 x
n
i , 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 ν(x1) = ν(x2) = · · · = ν(xn) and at least one
xi is in R¯ν .
5
a⊙ (b⊕ c) =
a⊙ b, b ≻ c,
a⊙ bν , b = c,
a⊙ c, c ≻ b,
and
(a⊙ b)⊕ (a⊙ c) =
a⊙ b, b ≻ c,
(a⊙ b)ν , b = c,
a⊙ c, c ≻ b,
and compare the evaluations with respect to the different ordering of the involved arguments.
When elements of both R and Rν are involved, use the above specification together with Axiom
1.4; for example,
aν ⊙ (b ⊕ c) = (a⊙ (b⊕ c))ν =
((a⊙ b)⊕ (a⊙ c))ν =
(a⊙ b)ν ⊕ (a⊙ c)ν = (a⊙ bν)⊕ (a⊙ cν) .
Zero: By definition 0
T
:= −∞ 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 1
T
:= 0 is the multiplicative identity of T.
Theorem 1.8. The set T equipped with the addition ⊕ and the multiplication ⊙ is a (non-
idempotent) commutative semiring, (Rν ,⊕) is an additive semigroup, and (R,⊙) and (Rν ,⊙) are
multiplicative semigroups.
Remark 1.9. In the view of Axiom 1.4, ν is realized as the onto order preserving projection
(1.3) ν : (T,⊕,⊙) −→ (R¯ν ,⊕,⊙) ,
where ν : a 7→ aν , ν : aν 7→ aν , and ν : −∞ 7→ −∞. Then, ν is a semiring homomorphism and
we write xν for the image of x ∈ T in R¯ν , where ν is is the identity for each x ∈ R¯ν . Accordingly,
call aν the ν-value of a. Given x, y ∈ T, we say that x is greater than y, or maximal, up to ν if
xν ≻ yν , similarly, when xν = yν we say that x and y are equal up to ν .
Writing xn for the tropical product x⊙ x⊙ · · · ⊙ x of n factors we have:
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, we have x⊕y ∈ R¯ν ,
xn ⊕ yn ∈ R¯ν , and xn ⊕ yn = xn = (x⊕ y)n.
Corollary 1.11. (
⊕s
i=1 xi)
n =
⊕s
i=1 x
n
i , 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 ν(x1) = ν(x2) = · · · = ν(xn) and at least one
xi is in R¯ν .
5
Page 6
1.4. Tropical arithmetics and tropicalization. The informal term tropicalization is used to
describe a map, based on a real valuation, of objects defined over a non-Archimedean field K with
real valuation 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 a polynomial in n
variables in K[λ1, . . . , λn] is mapped to a tropical polynomial in n variables in R¯[λ1, . . . , λn], which
we recall determines an affine piecewise linear function.
Let K be an algebraically closed field with a real non-Archimedean valuation
(1.4) V al : (K,+, · ) −→ (R¯,max,+) ;
for example, assume K is the field of locally convergent complex Puiseux series, of the form
f(t) =
∑
a∈R
cata, ca ∈ C ,
where R ⊂ Q is bounded from below and the elements of R have a bounded denominator. Then,
(1.5) V al(f) =
{
−min{a ∈ R : ca 6= 0}, f ∈ K[λ1, . . . , λn] \ 0 ;
−∞, f = 0 ,
is a real valuation satisfying the rules of being non-Archimedean,
(1.6)
(i) V al(f · g) = V al(f) + V al(g) ,
(ii) V al(f + g) ≤ max{V al(f), V al(g)} .
(Note that V al is not a homomorphism, since it does not preserve associativity.) Thus, in the sense
of tropicalization, the arithmetic operations of K are replaced with the correspondence: · 7→ +
and + 7→ max .
Remark 1.13. Taking f, g ∈ K with V al(f) = V al(g) = a, then V al(f + g) can be any point of
the ray [−∞, a]. These cases provide the motivation for the use of (T,⊕,⊙) as the target of V al
that allows to distinguish between the cases in which Formula (1.6)(ii) is interpreted as equality
and the cases it is inequality.
In order to realize (T,⊕,⊙) as the target of V al, to each point aν ∈ Rν we assign the ray
Paν := [−∞, a] and to each x ∈ R¯ we assign the singleton Pa := {a}, in particular P−∞ := {−∞};
therefore x ∈ Px for each x ∈ T. With this construction we obtain the inclusions:
(1.7) P−∞, Pa ⊂ Paν , Paν ⊂ Pbν ⇐⇒ a ≺ b, ∀a, b ∈ R .
(Recall that two series in K that are vanished in order 1 must vanished on order at least 1; the
inclusions (1.7) address this property.)
Let G(T) := {Px : x ∈ T}, then V al(f) ∈ Px for some Px ∈ G(T) which clearly needs not be
unique. Accordingly, for each pair f ∈ K and x ∈ T we define the relation
(1.8) V al(f) ∈ Px or V al(f) /∈ Px
determined by the inclusion of V al(f) in Px.
Theorem 1.14. Formula (1.8) yields a homomorphism; that is, for any f, g ∈ K with V al(f) ∈ Px
and V al(g) ∈ Py we have V al(f · g) ∈ Px⊙y and V al(f + g) ∈ Px⊕y.
Proof. Suppose V al(f) = a, V al(g) = b. Then, since x ∈ Px for each x ∈ T,
V al(f · g) = V al(f) + V al(g) = a⊙ b ∈ Pa⊙b ,
and V al(f · 0) = V al(f) + V al(0) = a⊙ (−∞) ∈ P−∞ . For the additive relation, write
V al(f + g) ≤ max{V al(f), V al(g)} =
max{a, b} =
a ∈ Pa = Pa⊕b a > b,
a ∈ Pa ⊂ Paν = Pa⊕a a = b,
b ∈ Pb = Pa⊕b b > a,
and use the inclusion Pa ⊂ Paν , cf. (1.7). The case of V al(f + 0) is trivial.
6
describe a map, based on a real valuation, of objects defined over a non-Archimedean field K with
real valuation 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 a polynomial in n
variables in K[λ1, . . . , λn] is mapped to a tropical polynomial in n variables in R¯[λ1, . . . , λn], which
we recall determines an affine piecewise linear function.
Let K be an algebraically closed field with a real non-Archimedean valuation
(1.4) V al : (K,+, · ) −→ (R¯,max,+) ;
for example, assume K is the field of locally convergent complex Puiseux series, of the form
f(t) =
∑
a∈R
cata, ca ∈ C ,
where R ⊂ Q is bounded from below and the elements of R have a bounded denominator. Then,
(1.5) V al(f) =
{
−min{a ∈ R : ca 6= 0}, f ∈ K[λ1, . . . , λn] \ 0 ;
−∞, f = 0 ,
is a real valuation satisfying the rules of being non-Archimedean,
(1.6)
(i) V al(f · g) = V al(f) + V al(g) ,
(ii) V al(f + g) ≤ max{V al(f), V al(g)} .
(Note that V al is not a homomorphism, since it does not preserve associativity.) Thus, in the sense
of tropicalization, the arithmetic operations of K are replaced with the correspondence: · 7→ +
and + 7→ max .
Remark 1.13. Taking f, g ∈ K with V al(f) = V al(g) = a, then V al(f + g) can be any point of
the ray [−∞, a]. These cases provide the motivation for the use of (T,⊕,⊙) as the target of V al
that allows to distinguish between the cases in which Formula (1.6)(ii) is interpreted as equality
and the cases it is inequality.
In order to realize (T,⊕,⊙) as the target of V al, to each point aν ∈ Rν we assign the ray
Paν := [−∞, a] and to each x ∈ R¯ we assign the singleton Pa := {a}, in particular P−∞ := {−∞};
therefore x ∈ Px for each x ∈ T. With this construction we obtain the inclusions:
(1.7) P−∞, Pa ⊂ Paν , Paν ⊂ Pbν ⇐⇒ a ≺ b, ∀a, b ∈ R .
(Recall that two series in K that are vanished in order 1 must vanished on order at least 1; the
inclusions (1.7) address this property.)
Let G(T) := {Px : x ∈ T}, then V al(f) ∈ Px for some Px ∈ G(T) which clearly needs not be
unique. Accordingly, for each pair f ∈ K and x ∈ T we define the relation
(1.8) V al(f) ∈ Px or V al(f) /∈ Px
determined by the inclusion of V al(f) in Px.
Theorem 1.14. Formula (1.8) yields a homomorphism; that is, for any f, g ∈ K with V al(f) ∈ Px
and V al(g) ∈ Py we have V al(f · g) ∈ Px⊙y and V al(f + g) ∈ Px⊕y.
Proof. Suppose V al(f) = a, V al(g) = b. Then, since x ∈ Px for each x ∈ T,
V al(f · g) = V al(f) + V al(g) = a⊙ b ∈ Pa⊙b ,
and V al(f · 0) = V al(f) + V al(0) = a⊙ (−∞) ∈ P−∞ . For the additive relation, write
V al(f + g) ≤ max{V al(f), V al(g)} =
max{a, b} =
a ∈ Pa = Pa⊕b a > b,
a ∈ Pa ⊂ Paν = Pa⊕a a = b,
b ∈ Pb = Pa⊕b b > a,
and use the inclusion Pa ⊂ Paν , cf. (1.7). The case of V al(f + 0) is trivial.
6
Page 7
1.5. The relation to the max-plus arithmetic. The structure of (T,⊕,⊙) provides a much
richer structure, generalizing both the max-plus semiring and the one suggested in Remark 1.1,
and achieves the best of both worlds.
Lemma 1.15. The map
(1.9) pi : (T,⊕,⊙) −→ (R¯,max,+ ) ,
pi : aν 7→ a, pi : a 7→ a, and pi : −∞ 7→ −∞, is a semiring epimorphism.
Proof. Clearly, pi is onto. Assume pi(x) = a and pi(y) = b, where x, y ∈ T, then pi(x ⊕ y) =
max{a, b} = max{pi(x), pi(y)} and pi(x ⊙ y) = a + b = pi(x) + pi(y).
On the other hand one can also define:
Lemma 1.16. The map
(1.10) θ : (R¯,max,+ ) −→ (R¯ν ,⊕,⊙),
θ : a 7→ aν and θ : −∞ 7→ −∞, is a semiring isomorphism that embeds (R¯,max,+ ) in (T,⊕,⊙).
Proof. Take a, b ∈ R¯, then θ(max{a, b}) = (max{a, b})ν = aν ⊕ bν = θ(a) ⊕ θ(b), and θ(a + b) =
(a + b)ν = aν ⊙ bν = θ(a)⊙ θ(b). R¯ν ⊂ T, so θ embeds (R¯,max,+ ) in (T,⊕,⊙).
Corollary 1.17. Categorically, by Remark 1.9, Lemma 1.15, and Lemma 1.16, the diagram
(T,⊕,⊙) (R¯,max,+ )
(R¯ν ,⊕,⊙)
pi
ν θ
-
j
?
commutes.
Corollary 1.17 displays (T,⊕,⊙) as a generalization of (R¯,max,+ ) which is endowed with
a richer structure in the sense that it encodes an indication about the additive multiplicity of
elements in R. Namely, since a ⊕ a = aν and a ⊕ aν = aν , aν can be realized as a point with
additive multiplicity > 1. (Clearly, computations for (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
(1.11) φ : (T,⊕,⊙) −→ (R¯, “max ”,+ ),
φ : a 7→ a, φ : aν 7→ −∞, and φ : −∞ 7→ −∞; but, since (R¯, “max ”,+ ) is not associative, φ is
not a homomorphism.
1.6. Geometric view. Let us remind that one of our goals was to obtain a semiring structure
that enables 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, given a tropical polynomial
f ∈ T[λ1, . . . , λn] we define the tropical algebraic set of the corresponding function f˜ to be
Z(f˜) = {(x1, . . . , xn) ∈ T(n) : f˜(x1, . . . , xn) ∈ R¯ν}.
Therefore, the corner locus of f ∈ R¯[λ1, . . . , λn] over (R¯,max,+ ) is just the restriction of Z(f˜),
considered as a polynomial over (T,⊕,⊙), to the real points, i.e. Z(f˜) ∩ 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 Figure 1. Restricting the domain to R only, over (T,⊕,⊙), the
image of the corner locus, which contains the single point a, is distinguished and is now mapped
to Rν .
The study of polynomial algebras and tropical algebraic sets over (T,⊕,⊙) will be treated in a
forthcoming paper.
7
richer structure, generalizing both the max-plus semiring and the one suggested in Remark 1.1,
and achieves the best of both worlds.
Lemma 1.15. The map
(1.9) pi : (T,⊕,⊙) −→ (R¯,max,+ ) ,
pi : aν 7→ a, pi : a 7→ a, and pi : −∞ 7→ −∞, is a semiring epimorphism.
Proof. Clearly, pi is onto. Assume pi(x) = a and pi(y) = b, where x, y ∈ T, then pi(x ⊕ y) =
max{a, b} = max{pi(x), pi(y)} and pi(x ⊙ y) = a + b = pi(x) + pi(y).
On the other hand one can also define:
Lemma 1.16. The map
(1.10) θ : (R¯,max,+ ) −→ (R¯ν ,⊕,⊙),
θ : a 7→ aν and θ : −∞ 7→ −∞, is a semiring isomorphism that embeds (R¯,max,+ ) in (T,⊕,⊙).
Proof. Take a, b ∈ R¯, then θ(max{a, b}) = (max{a, b})ν = aν ⊕ bν = θ(a) ⊕ θ(b), and θ(a + b) =
(a + b)ν = aν ⊙ bν = θ(a)⊙ θ(b). R¯ν ⊂ T, so θ embeds (R¯,max,+ ) in (T,⊕,⊙).
Corollary 1.17. Categorically, by Remark 1.9, Lemma 1.15, and Lemma 1.16, the diagram
(T,⊕,⊙) (R¯,max,+ )
(R¯ν ,⊕,⊙)
pi
ν θ
-
j
?
commutes.
Corollary 1.17 displays (T,⊕,⊙) as a generalization of (R¯,max,+ ) which is endowed with
a richer structure in the sense that it encodes an indication about the additive multiplicity of
elements in R. Namely, since a ⊕ a = aν and a ⊕ aν = aν , aν can be realized as a point with
additive multiplicity > 1. (Clearly, computations for (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
(1.11) φ : (T,⊕,⊙) −→ (R¯, “max ”,+ ),
φ : a 7→ a, φ : aν 7→ −∞, and φ : −∞ 7→ −∞; but, since (R¯, “max ”,+ ) is not associative, φ is
not a homomorphism.
1.6. Geometric view. Let us remind that one of our goals was to obtain a semiring structure
that enables 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, given a tropical polynomial
f ∈ T[λ1, . . . , λn] we define the tropical algebraic set of the corresponding function f˜ to be
Z(f˜) = {(x1, . . . , xn) ∈ T(n) : f˜(x1, . . . , xn) ∈ R¯ν}.
Therefore, the corner locus of f ∈ R¯[λ1, . . . , λn] over (R¯,max,+ ) is just the restriction of Z(f˜),
considered as a polynomial over (T,⊕,⊙), to the real points, i.e. Z(f˜) ∩ 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 Figure 1. Restricting the domain to R only, over (T,⊕,⊙), the
image of the corner locus, which contains the single point a, is distinguished and is now mapped
to Rν .
The study of polynomial algebras and tropical algebraic sets over (T,⊕,⊙) will be treated in a
forthcoming paper.
7
Page 8
x ∈ Rx ∈ T
f˜(x)f˜(x)
a
◦◦
• •
aaν −∞
(T,⊕,⊙) (R¯,max,+ )
Figure 1. The graph of the linear function f˜(x) = x + a over (T,⊕,⊙), on left
hand side, and the corresponding function f˜(x) = max{x, a} over (R¯,max,+ ), on
right hand side.
2. Matrix Algebra
Our forthcoming study is dedicated to introducing the fundamentals of the matrix algebra over
(T,⊕,⊙) whose operations of are typically combinatorial. Yet, developing an algebraic theory,
analogous to classical theory of matrix algebra over fields, with a view to combinatorics, is our
main goal. This goal is supported by the connections to graph theory [9], the theory of automata
[10], and semiring theory [5].
Notations: For the rest of the paper, assuming the nuances of the different arithmetics are
already familiar, we write xy for the product x⊙y, xy for 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 × n matrices with entries in R, where addition and multiplication are induced from R as in
the familiar matrix construction. Accordingly, we define the semiring of tropical matrices Mn(T)
over (T,⊕,⊙), whose unit is the matrix
(2.1) I =
0 . . . −∞
...
. . .
...
−∞ . . . 0
and whose zero matrix is Z = (−∞)I; therefore, Mn(T) is also a multiplicative monied. We write
A = (aij) for a tropical matrices A ∈ Mn(T) and denote the entries of A as aij . Since T is a
commutative semiring, xA = Ax for any x ∈ T and A ∈ Mn(T).
As in the familiar way, we define the transpose of A = (aij) to be At = (aji), and have the
relation
Proposition 2.1. (AB) t = BtAt.
(The proof is standard by the commutativity and the associativity of ⊕ and ⊙ over T.)
The minor Aij is obtained by deleting the i row and j column of A. We define the tropical
determinant to be
(2.2) |A| =
⊕
σ∈Sn
(
a1σ(1) · · ·anσ(n)
)
,
where Sn is the set of all the permutations on {1, . . . , n}. Equivalently, |A| can be written in terms
of minors as
(2.3) |A| =
⊕
j
aioj |Aioj |,
8
Page 9
for some fixed index io. Indeed, in the classical sense, since parity of indices’ sums are not involved
in Formula (2.2), the tropical determinant is a permanent, which makes the tropical determinant
a pure combinatorial function. The adjoint matrix Adj(A) of A = (aij) is defined as the matrix
(a′ij)
t where a′ij = |Aij |.
Observation 2.2. The tropical determinant has the following properties:
(1) Transposition and reordering of rows or columns leave the determinant unchanged;
(2) The determinant is linear with respect to scalar multiplication of any given row or column.
2.2. Regularity of matrices. Using the special structure of (T,⊕,⊙), the algebraic formulation
of combinatorial properties becomes possible.
Definition 2.3. A matrix A ∈ Mn(T) is said to be tropically singular, or singular, for short,
whenever |A| ∈ R¯ν , otherwise A is called tropically regular, or regular, for short.
In particular, when two or more different permutations, σ ∈ Sn, achieve the ν-value of |A|
simultaneously, or the permutation that reaches the ν-value of |A| involves an entry in R¯ν , then A
is singular.
Remark 2.4. Despite some classical properties hold for the tropical determinant, cf. Observa-
tion 2.2, the familiar relation |AB| = |A||B| does not hold true on our setting; for example, take
the matrix
(2.4) A =
(
1 1
2 3
)
with A2 = AA =
(
3 4
5 6
)
,
then, |A| = 4 and |A||A| = 8, while |A2| = 9ν . In the view of tropicalization, which ignores
signs, the determinant of a matrix over K is assigned to the permanent of a matrix in Mn(T); this
explains the tropical situation in which the product of two regular 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 direct computation. Assume
the two first columns of A are identical, and expand |A| in terms of minors along the first row,
that is |A| = ⊕i a1i|A1i|. Since a11 = a12 and A11 = A12, then a11|A11| = a12|A12|, and so
a11|A11|⊕a12|A12| ∈ R¯ν . By the induction hypothesis, for any i > 2, A1i is a matrix with identical
columns, and is singular, that is |A1i| ∈ R¯ν for all i > 2. When adding all together, a1i|A1i| ∈ R¯ν
for all i = 1, . . . , n, and thus |A| ∈ Rν .
Theorem 2.6. If A and B are regular matrices and their product AB is also regular, then |AB| =
|A||B|. When either A or B is singular, then AB is also singular.
Proof. Let Sn be the set of all the permutations on N = {1, . . . , n}, and let Fn = {N −→ N} be
the set of all maps from N to itself, in particular Sn ⊂ Fn. Denoting the entries of AB by (ab)ij ,
we write the determinant |AB| in the form of Formula (2.2) as:
|AB| =
⊕
σ∈Sn
⊙
i
(ab)iσ(i) =
⊕
σ∈Sn
⊙
i
(
⊕
k
(aikbkσ(i))
)
=
⊕
σ∈Sn
(
(a11b1σ(1) ⊕ · · · ⊕ a1nbnσ(1)) · · · (an1b1σ(n) ⊕ · · · ⊕ annbnσ(n))
)
=
(*)
⊕
σ∈Sn
⊕
µ∈Fn
(
⊙
i
(
aiµ(i)bµ(i)σ(i)
)
)
=
⊕
σ∈Sn
⊕
µ∈Fn
(
⊙
i
aiµ(i)
⊙
i
bµ(i)σ(i)
)
.
By the structure of the left hand side of (∗), we can see that the value of |AB| is obtained when
both
⊙
i aiµ(i) and
⊙
i bµ(i)σ(i) attain their maximal evaluation at the same time. We show that
this is possible. Namely both reach their maximal evaluation on the same µ, which we denote by
µo; the corresponding σ ∈ Sn is then denoted by σo. Note that when |AB| ∈ R there must be
exactly one pair, µo and σo; otherwise, by definition, AB would not be regular.
9
in Formula (2.2), the tropical determinant is a permanent, which makes the tropical determinant
a pure combinatorial function. The adjoint matrix Adj(A) of A = (aij) is defined as the matrix
(a′ij)
t where a′ij = |Aij |.
Observation 2.2. The tropical determinant has the following properties:
(1) Transposition and reordering of rows or columns leave the determinant unchanged;
(2) The determinant is linear with respect to scalar multiplication of any given row or column.
2.2. Regularity of matrices. Using the special structure of (T,⊕,⊙), the algebraic formulation
of combinatorial properties becomes possible.
Definition 2.3. A matrix A ∈ Mn(T) is said to be tropically singular, or singular, for short,
whenever |A| ∈ R¯ν , otherwise A is called tropically regular, or regular, for short.
In particular, when two or more different permutations, σ ∈ Sn, achieve the ν-value of |A|
simultaneously, or the permutation that reaches the ν-value of |A| involves an entry in R¯ν , then A
is singular.
Remark 2.4. Despite some classical properties hold for the tropical determinant, cf. Observa-
tion 2.2, the familiar relation |AB| = |A||B| does not hold true on our setting; for example, take
the matrix
(2.4) A =
(
1 1
2 3
)
with A2 = AA =
(
3 4
5 6
)
,
then, |A| = 4 and |A||A| = 8, while |A2| = 9ν . In the view of tropicalization, which ignores
signs, the determinant of a matrix over K is assigned to the permanent of a matrix in Mn(T); this
explains the tropical situation in which the product of two regular 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 direct computation. Assume
the two first columns of A are identical, and expand |A| in terms of minors along the first row,
that is |A| = ⊕i a1i|A1i|. Since a11 = a12 and A11 = A12, then a11|A11| = a12|A12|, and so
a11|A11|⊕a12|A12| ∈ R¯ν . By the induction hypothesis, for any i > 2, A1i is a matrix with identical
columns, and is singular, that is |A1i| ∈ R¯ν for all i > 2. When adding all together, a1i|A1i| ∈ R¯ν
for all i = 1, . . . , n, and thus |A| ∈ Rν .
Theorem 2.6. If A and B are regular matrices and their product AB is also regular, then |AB| =
|A||B|. When either A or B is singular, then AB is also singular.
Proof. Let Sn be the set of all the permutations on N = {1, . . . , n}, and let Fn = {N −→ N} be
the set of all maps from N to itself, in particular Sn ⊂ Fn. Denoting the entries of AB by (ab)ij ,
we write the determinant |AB| in the form of Formula (2.2) as:
|AB| =
⊕
σ∈Sn
⊙
i
(ab)iσ(i) =
⊕
σ∈Sn
⊙
i
(
⊕
k
(aikbkσ(i))
)
=
⊕
σ∈Sn
(
(a11b1σ(1) ⊕ · · · ⊕ a1nbnσ(1)) · · · (an1b1σ(n) ⊕ · · · ⊕ annbnσ(n))
)
=
(*)
⊕
σ∈Sn
⊕
µ∈Fn
(
⊙
i
(
aiµ(i)bµ(i)σ(i)
)
)
=
⊕
σ∈Sn
⊕
µ∈Fn
(
⊙
i
aiµ(i)
⊙
i
bµ(i)σ(i)
)
.
By the structure of the left hand side of (∗), we can see that the value of |AB| is obtained when
both
⊙
i aiµ(i) and
⊙
i bµ(i)σ(i) attain their maximal evaluation at the same time. We show that
this is possible. Namely both reach their maximal evaluation on the same µ, which we denote by
µo; the corresponding σ ∈ Sn is then denoted by σo. Note that when |AB| ∈ R there must be
exactly one pair, µo and σo; otherwise, by definition, AB would not be regular.
9
Page 10
Case I: Suppose µo ∈ Sn is a permutation which maximizes
⊙
i aiµ(i). We show that there is
also a permutation σo ∈ Sn that maximizes
⊙
i bµ(i)σ(i) for µo. Assume |AB| ∈ R, with σt ∈ Sn
maximizes
⊙
j bjσt(j). Generally speaking, for any given µ ∈ Sn and σt ∈ Sn, there exits σ ∈ Sn
which makes the diagram
N[j] N[i]-
µ
σt
R
N
?
σ
commutative, where we use the notation [ · ] to indicate the appropriate indices. Accordingly,
choosing σo ∈ Sn for which σo ◦µo = σt, we obtain
⊙
i bµo(i)σo(i) =
⊙
j bjσt(j); in this case the two
components of (∗) reach their maximum simultaneously and we can write:
(∗) =
(
⊙
i
aiµo(i)
)
⊙
j
bjνo(j)
=
⊕
µ∈Sn
⊙
i
aiµ(i)
⊕
ν∈Sn
⊙
j
bjν(j)
= |A||B| .
When A is singular, there are at least two different µ1, µ2 ∈ Sn that attain the ν-value of |A| or
a single µ ∈ Sn that involves a non-real entry. The latter case is obvious, since (∗) has a non-real
multiplier and thus |AB| ∈ R¯ν . Suppose σ1, σ2 ∈ Sn are two permutations satisfying νo = σl ◦ µl,
l = 1, 2, then
(∗) =
⊙
i
aiµ1(i)
⊙
i
bµ1(i)σ1(i) =
⊙
i
aiµ2(i)
⊙
i
bµ2(i)σ2(i) ,
and hence |AB| ∈ R¯ν .
Case II: Suppose µo ∈ Fn \ Sn, |AB| ∈ R, and let σo ∈ Sn be the corresponding permutation
which maximizes the product
(**)
⊙
i
(
aiµ(i)bµ(i)σ(i)
)
=
⊙
i
aiµ(i)
⊙
i
bµ(i)σ(i) .
in (∗). In particular, there is only one such pair, µo and σo, for otherwise AB would not be regular.
Since µo /∈ Sn, there are at least two indices i1 6= i2 with µo(i1) = µo(i2) = ko. Let h1 := σo(i1)
and h2 := σo(i2); then h1 6= h2, since σo ∈ Sn. Subject to µo and σo, (∗∗) can be rewritten as,
(
⊙
i
aiµo(i)
)(
⊙
i
bµo(i)σo(i)
)
=
(
⊙
i
aiµo(i)
)
(bµo(i1),σo(i1)bµo(i2)σo(i2))
⊙
i6=i1i2
bµo(i)σo(i)
=
(***)
(
⊙
i
ai,µo(i)
)
(bkoh1bkoh2)
⊙
i6=i1i2
bµo(i)σo(i)
.
Denote by σ˜o ∈ Sn the permutation obtained by switching between the images of i1 and i2 in
σo, while all other correspondences remain as they are; explicitly, σ˜o(i1) = h2, σ˜o(i2) = h1, and
σ˜o(i) = σo(i), for all i 6= i1, i2. With respect to σo, we have the following combinatorial situation:
σo(i1)
= h1
σo(i2)
= h2
...
...
µo(i1)=
µo(i2)=
ko · · · ∗ · · · ∗ · · ·
...
...
=σ˜o(i2) =σ˜o(i1)
10
⊙
i aiµ(i). We show that there is
also a permutation σo ∈ Sn that maximizes
⊙
i bµ(i)σ(i) for µo. Assume |AB| ∈ R, with σt ∈ Sn
maximizes
⊙
j bjσt(j). Generally speaking, for any given µ ∈ Sn and σt ∈ Sn, there exits σ ∈ Sn
which makes the diagram
N[j] N[i]-
µ
σt
R
N
?
σ
commutative, where we use the notation [ · ] to indicate the appropriate indices. Accordingly,
choosing σo ∈ Sn for which σo ◦µo = σt, we obtain
⊙
i bµo(i)σo(i) =
⊙
j bjσt(j); in this case the two
components of (∗) reach their maximum simultaneously and we can write:
(∗) =
(
⊙
i
aiµo(i)
)
⊙
j
bjνo(j)
=
⊕
µ∈Sn
⊙
i
aiµ(i)
⊕
ν∈Sn
⊙
j
bjν(j)
= |A||B| .
When A is singular, there are at least two different µ1, µ2 ∈ Sn that attain the ν-value of |A| or
a single µ ∈ Sn that involves a non-real entry. The latter case is obvious, since (∗) has a non-real
multiplier and thus |AB| ∈ R¯ν . Suppose σ1, σ2 ∈ Sn are two permutations satisfying νo = σl ◦ µl,
l = 1, 2, then
(∗) =
⊙
i
aiµ1(i)
⊙
i
bµ1(i)σ1(i) =
⊙
i
aiµ2(i)
⊙
i
bµ2(i)σ2(i) ,
and hence |AB| ∈ R¯ν .
Case II: Suppose µo ∈ Fn \ Sn, |AB| ∈ R, and let σo ∈ Sn be the corresponding permutation
which maximizes the product
(**)
⊙
i
(
aiµ(i)bµ(i)σ(i)
)
=
⊙
i
aiµ(i)
⊙
i
bµ(i)σ(i) .
in (∗). In particular, there is only one such pair, µo and σo, for otherwise AB would not be regular.
Since µo /∈ Sn, there are at least two indices i1 6= i2 with µo(i1) = µo(i2) = ko. Let h1 := σo(i1)
and h2 := σo(i2); then h1 6= h2, since σo ∈ Sn. Subject to µo and σo, (∗∗) can be rewritten as,
(
⊙
i
aiµo(i)
)(
⊙
i
bµo(i)σo(i)
)
=
(
⊙
i
aiµo(i)
)
(bµo(i1),σo(i1)bµo(i2)σo(i2))
⊙
i6=i1i2
bµo(i)σo(i)
=
(***)
(
⊙
i
ai,µo(i)
)
(bkoh1bkoh2)
⊙
i6=i1i2
bµo(i)σo(i)
.
Denote by σ˜o ∈ Sn the permutation obtained by switching between the images of i1 and i2 in
σo, while all other correspondences remain as they are; explicitly, σ˜o(i1) = h2, σ˜o(i2) = h1, and
σ˜o(i) = σo(i), for all i 6= i1, i2. With respect to σo, we have the following combinatorial situation:
σo(i1)
= h1
σo(i2)
= h2
...
...
µo(i1)=
µo(i2)=
ko · · · ∗ · · · ∗ · · ·
...
...
=σ˜o(i2) =σ˜o(i1)
10
Page 11
(The diagram helps us to understand the modification of µo.) Using σ˜ we expand (∗ ∗ ∗) further,
(∗ ∗ ∗) =
(
⊙
i
aiµo(i)
)
(bµo(i2)σ˜o(i2)bµo(i1)σ˜o(i1))
⊙
i6=i1i2
bµo(i)σ˜o(i)
=
(
⊙
i
aiµo(i)
)(
⊙
i
bµo(i)σ˜o(i)
)
,
which means that σ˜o 6= σo also attain the ν-value of |AB| and thus, |AB| ∈ R¯ν . This contradicts
the assumption that µo /∈ Sn, so µo ∈ Sn, and this case has already been discussed before.
Example 2.7. Take the matrices
A =
(
1 2
2 3
)
and B =
(
3 1
0 2
)
, then AB =
(
4 4
5 5
)
.
A is singular with |A| = 4ν , B is regular with |B| = 5, and AB is singular with |AB| = 9ν . On the
other hand, B2 =
(
6 4
3 4
)
is regular with |B2| = 10, so |B||B| = |B2|.
3. Invertibility of Matrices
We introduce a new notion of semigroup invertibility, and present it for the matrix monoid
Mn(T); this type of invertibility can be adopted to any abstract semigroup having a distinguished
subset. Although our framework is typically combinatorial, 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 S be semigroup, and let U ⊂ S be a proper subset with the property 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 ∈ S is said to be pseudo invariable if there is y ∈ S for which xy ∈ U
and yx ∈ U , in particular all the members of U are pseudo invertible. When U consists of all
idempotents elements of S, the pseudo invertibility is then called E-denseness [10]. A monoid is
called E-dense if all of its elements are E-denseness.
To emphasize, for the purpose of pseudo invertibility, U needs not be closed under the law of
S. The notion of E-denseness is already known in literature, while the weaker version of pseudo
invertibility is new.
To apply the notion of pseudo invariability to Mn(T), viewed as monoid, we define a pseudo
unit matrix to be a regular matrix of the form
(3.1) I˜ =
0 . . . ινij
...
. . .
...
ινji . . . 0
.
that is ιij ∈ R¯ν for all i 6= j, and ιii = 0, for each i = 1, . . . , n. Since I˜ is regular we necessarily
have |I˜| = 0, and in particular the unit matrix I, cf. (2.1), is also a pseudo unit. We define the
distinguished subset Un(T) ⊂ Mn(T) to be
(3.2) Un(T) =
{
I˜ : I˜ is a pseudo unit matrix
}
,
Therefore, I ∈ Un(T) and hence II˜ = I˜I = I˜, for each I˜ ∈ Un(T), which makes Un(T) a
distinguished subset satisfies the condition of Definition 3.1.
Correspondingly, we define the distinguished subset U idemn (T) ⊂ Mn(T) to be
(3.3) U idmn (T) =
{
I˜ : I˜ is an idempotent pseudo unit matrix
}
.
11
(∗ ∗ ∗) =
(
⊙
i
aiµo(i)
)
(bµo(i2)σ˜o(i2)bµo(i1)σ˜o(i1))
⊙
i6=i1i2
bµo(i)σ˜o(i)
=
(
⊙
i
aiµo(i)
)(
⊙
i
bµo(i)σ˜o(i)
)
,
which means that σ˜o 6= σo also attain the ν-value of |AB| and thus, |AB| ∈ R¯ν . This contradicts
the assumption that µo /∈ Sn, so µo ∈ Sn, and this case has already been discussed before.
Example 2.7. Take the matrices
A =
(
1 2
2 3
)
and B =
(
3 1
0 2
)
, then AB =
(
4 4
5 5
)
.
A is singular with |A| = 4ν , B is regular with |B| = 5, and AB is singular with |AB| = 9ν . On the
other hand, B2 =
(
6 4
3 4
)
is regular with |B2| = 10, so |B||B| = |B2|.
3. Invertibility of Matrices
We introduce a new notion of semigroup invertibility, and present it for the matrix monoid
Mn(T); this type of invertibility can be adopted to any abstract semigroup having a distinguished
subset. Although our framework is typically combinatorial, 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 S be semigroup, and let U ⊂ S be a proper subset with the property 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 ∈ S is said to be pseudo invariable if there is y ∈ S for which xy ∈ U
and yx ∈ U , in particular all the members of U are pseudo invertible. When U consists of all
idempotents elements of S, the pseudo invertibility is then called E-denseness [10]. A monoid is
called E-dense if all of its elements are E-denseness.
To emphasize, for the purpose of pseudo invertibility, U needs not be closed under the law of
S. The notion of E-denseness is already known in literature, while the weaker version of pseudo
invertibility is new.
To apply the notion of pseudo invariability to Mn(T), viewed as monoid, we define a pseudo
unit matrix to be a regular matrix of the form
(3.1) I˜ =
0 . . . ινij
...
. . .
...
ινji . . . 0
.
that is ιij ∈ R¯ν for all i 6= j, and ιii = 0, for each i = 1, . . . , n. Since I˜ is regular we necessarily
have |I˜| = 0, and in particular the unit matrix I, cf. (2.1), is also a pseudo unit. We define the
distinguished subset Un(T) ⊂ Mn(T) to be
(3.2) Un(T) =
{
I˜ : I˜ is a pseudo unit matrix
}
,
Therefore, I ∈ Un(T) and hence II˜ = I˜I = I˜, for each I˜ ∈ Un(T), which makes Un(T) a
distinguished subset satisfies the condition of Definition 3.1.
Correspondingly, we define the distinguished subset U idemn (T) ⊂ Mn(T) to be
(3.3) U idmn (T) =
{
I˜ : I˜ is an idempotent pseudo unit matrix
}
.
11
Page 12
Remark 3.2. It easy to show that any I˜ ∈ U2(T) is idempotent. For n > 2, not all of the pseudo
units are idempotents; for example, take the triangular matrix
I˜ =
0 aν bν
−∞ 0 cν
−∞ −∞ 0
,
with aνcν ≻ bν .
Using Un(T) we explicitly define pseudo invertibility on Mn(T):
Definition 3.3. A matrix A ∈ Mn(T) is said to be pseudo invertible if there exits a matrix
B ∈ Mn(T) such that AB ∈ Un(T) and BA ∈ Un(T). If A is pseudo invertible, then we call B a
pseudo inverse matrix of A and denote it as A▽.
We use the notation of A▽ since the pseudo matrix needs not be unique; moreover, in our
setting AA▽ is not necessarily equal to A▽A, and thus might be evaluated for different pseudo
units.
Example 3.4. Consider the following matrices:
A =
( 0 −2 −1
−2 0 (−3)ν
−1 (−3)ν 0
)
and A′ =
( 0 −2 −1
−2 0 −3
−1 −3 0
)
.
For these matrices we have, AA′ ∈ Un(T), A′A ∈ Un(T), and also AA ∈ Un(T). Namely, A has
at least two pseudo inverses.
Remark 3.5. For the case of M2(T), our notion of pseudo invertibility coincides with the notion
of general invertibility in a semigroup in the sense of Von-Neumann regularity [8], 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 and products of entries. Thus, obtaining the unit matrix I as products
of matrices is very restricted. On the other hand, when using (R¯, “max ”,+ ), as suggested in
Remark 1.1, multiplication is not associative, which makes implementation very difficult.
3.2. Theorem on tropical pseudo inverse matrix.
Theorem 3.7. A matrix A ∈ Mn(T) is pseudo invertible if and only if is tropically regular. In
case A is regular, A▽ can be defined as
A▽ = Adj(A)|A| .
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 the classical
sense. We write a−1 for 0a and a
m for the tropical product of a repeated m times (which
is just m · a in the usual sense).
(b) We use the notation Aih,jk for (Aij)hk, that is (h, k)-minor of the minor Aij , where h 6= i
and k 6= j with respect to the initial indices of A. Accordingly, |Aij | is written in terms of
minors as |Aij | =
⊕
k 6=j ahk|Aih,jk|, where h 6= i.
Proof. We prove only multiplication on right, AA▽ ∈ Un(T); the multiplication on left is proved
in the same way.
(⇐) Assume |A| ∈ R¯ν and at the same time there exists a pseudo inverseA▽. Then, by Theorem
2.6, |AA▽| ∈ R¯ν and their product is singular. Recalling that |I˜| = 0 for all I˜ ∈ Un(T), we have
AA▽ /∈ Un(T).
(⇒) We write A⋄ for the adjoint matrix Adj(A), for short, and denote the product AA⋄ by
B = (bij). Assuming A is regular, we need to prove that AA
⋄
|A| ∈ Un(T), or equivalently, that
AA⋄ = |A|I˜ for some I˜ ∈ Un(T). To prove this, we need to verify the following conditions:
12
units are idempotents; for example, take the triangular matrix
I˜ =
0 aν bν
−∞ 0 cν
−∞ −∞ 0
,
with aνcν ≻ bν .
Using Un(T) we explicitly define pseudo invertibility on Mn(T):
Definition 3.3. A matrix A ∈ Mn(T) is said to be pseudo invertible if there exits a matrix
B ∈ Mn(T) such that AB ∈ Un(T) and BA ∈ Un(T). If A is pseudo invertible, then we call B a
pseudo inverse matrix of A and denote it as A▽.
We use the notation of A▽ since the pseudo matrix needs not be unique; moreover, in our
setting AA▽ is not necessarily equal to A▽A, and thus might be evaluated for different pseudo
units.
Example 3.4. Consider the following matrices:
A =
( 0 −2 −1
−2 0 (−3)ν
−1 (−3)ν 0
)
and A′ =
( 0 −2 −1
−2 0 −3
−1 −3 0
)
.
For these matrices we have, AA′ ∈ Un(T), A′A ∈ Un(T), and also AA ∈ Un(T). Namely, A has
at least two pseudo inverses.
Remark 3.5. For the case of M2(T), our notion of pseudo invertibility coincides with the notion
of general invertibility in a semigroup in the sense of Von-Neumann regularity [8], 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 and products of entries. Thus, obtaining the unit matrix I as products
of matrices is very restricted. On the other hand, when using (R¯, “max ”,+ ), as suggested in
Remark 1.1, multiplication is not associative, which makes implementation very difficult.
3.2. Theorem on tropical pseudo inverse matrix.
Theorem 3.7. A matrix A ∈ Mn(T) is pseudo invertible if and only if is tropically regular. In
case A is regular, A▽ can be defined as
A▽ = Adj(A)|A| .
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 the classical
sense. We write a−1 for 0a and a
m for the tropical product of a repeated m times (which
is just m · a in the usual sense).
(b) We use the notation Aih,jk for (Aij)hk, that is (h, k)-minor of the minor Aij , where h 6= i
and k 6= j with respect to the initial indices of A. Accordingly, |Aij | is written in terms of
minors as |Aij | =
⊕
k 6=j ahk|Aih,jk|, where h 6= i.
Proof. We prove only multiplication on right, AA▽ ∈ Un(T); the multiplication on left is proved
in the same way.
(⇐) Assume |A| ∈ R¯ν and at the same time there exists a pseudo inverseA▽. Then, by Theorem
2.6, |AA▽| ∈ R¯ν and their product is singular. Recalling that |I˜| = 0 for all I˜ ∈ Un(T), we have
AA▽ /∈ Un(T).
(⇒) We write A⋄ for the adjoint matrix Adj(A), for short, and denote the product AA⋄ by
B = (bij). Assuming A is regular, we need to prove that AA
⋄
|A| ∈ Un(T), or equivalently, that
AA⋄ = |A|I˜ for some I˜ ∈ Un(T). To prove this, we need to verify the following conditions:
12
Page 13
(1) bii = |A| for each i;
(2) bij ∈ R¯ν , for any i 6= j;
(3) | B|A| | = 0.
Diagonal entries: When i = j,
(3.4) bii =
⊕
k
aika⋄ki =
⊕
k
aik|Aik| = |A|,
since this is just the expansion of |A| along row i (cf. Equation (2.3)).
Non-diagonal entries: For i 6= j,
(3.5) bij =
⊕
k
aika⋄kj =
⊕
k
aik|Ajk| ∈ R¯ν ,
since this is the expansion of the determinant of the matrix obtained from A by replacing row j
with a copy of row i, and which therefore has two identical row and is singular (Theorem 2.5).
Regularity of product: To prove | B|A| | = 0, we show equivalently that |B| = |A|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 |B| explicitly,
|B| =
⊕
σ∈Sn
(
⊙
i
biσ(i)
)
=
⊕
σ∈Sn
(
⊙
i
(
⊕
k
aik|Aσ(i)k|
))
=
⊕
σ∈Sn
((
a11|Aσ(1)1| ⊕ · · · ⊕ a1n|Aσ(1)n|
)
· · ·
(
an1|Aσ(n)1| ⊕ · · · ⊕ ann|Aσ(n)n|
))
=
(3.6)
⊕
σ∈Sn
⊕
µ∈Fn
⊙
i
(
aiµ(i)|Aσ(i)µ(i)|
)
.
Assume σo ∈ Sn and µo ∈ Mn achieve the ν-value of |B|. In case σo is the identity, by Equation
(3.4), bii = |A|, for each i, and thus,
(3.7)
⊙
i
biσo(i) =
⊙
i
bii = |A|n.
Otherwise, when σo is not the identity, we write
(3.8) c :=
⊙
i
(
aiµo(i)|Aσo(i)µo(i)|
)
,
for the product that reaches the ν-value of (3.6) and prove it always ≺ |A|n.
Case I: Assume µo ∈ Sn is a permutation, then Formula (3.8) can be reordered to the form
(*) a1µo(1)|A1µo(1)| · · · anµo(n)|Anµo(n)|.
If (∗) ≻ |A|n, then it must have at least one component ajµo(n)|Ajµo(n)| ≻ |A|, but this contradicts
the maximality of |A|. On the other hand, if all ajµo(n)|Ajµo(n)| = |A| we get a contradiction to
the regularity of |A|. Therefore, (∗) ≺ |A|n.
Case II: Assume µo ∈ Fn \ Sn, then there exist at least two indices i1 6= i2 for which µo(i1) =
µo(i2) = jo. We show the existence of a permutation µl ∈ Sn that reaches the same ν-value for
Formula (3.8) as µo reaches, the proof is then completed by Case I, applied to µl.
13
(2) bij ∈ R¯ν , for any i 6= j;
(3) | B|A| | = 0.
Diagonal entries: When i = j,
(3.4) bii =
⊕
k
aika⋄ki =
⊕
k
aik|Aik| = |A|,
since this is just the expansion of |A| along row i (cf. Equation (2.3)).
Non-diagonal entries: For i 6= j,
(3.5) bij =
⊕
k
aika⋄kj =
⊕
k
aik|Ajk| ∈ R¯ν ,
since this is the expansion of the determinant of the matrix obtained from A by replacing row j
with a copy of row i, and which therefore has two identical row and is singular (Theorem 2.5).
Regularity of product: To prove | B|A| | = 0, we show equivalently that |B| = |A|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 |B| explicitly,
|B| =
⊕
σ∈Sn
(
⊙
i
biσ(i)
)
=
⊕
σ∈Sn
(
⊙
i
(
⊕
k
aik|Aσ(i)k|
))
=
⊕
σ∈Sn
((
a11|Aσ(1)1| ⊕ · · · ⊕ a1n|Aσ(1)n|
)
· · ·
(
an1|Aσ(n)1| ⊕ · · · ⊕ ann|Aσ(n)n|
))
=
(3.6)
⊕
σ∈Sn
⊕
µ∈Fn
⊙
i
(
aiµ(i)|Aσ(i)µ(i)|
)
.
Assume σo ∈ Sn and µo ∈ Mn achieve the ν-value of |B|. In case σo is the identity, by Equation
(3.4), bii = |A|, for each i, and thus,
(3.7)
⊙
i
biσo(i) =
⊙
i
bii = |A|n.
Otherwise, when σo is not the identity, we write
(3.8) c :=
⊙
i
(
aiµo(i)|Aσo(i)µo(i)|
)
,
for the product that reaches the ν-value of (3.6) and prove it always ≺ |A|n.
Case I: Assume µo ∈ Sn is a permutation, then Formula (3.8) can be reordered to the form
(*) a1µo(1)|A1µo(1)| · · · anµo(n)|Anµo(n)|.
If (∗) ≻ |A|n, then it must have at least one component ajµo(n)|Ajµo(n)| ≻ |A|, but this contradicts
the maximality of |A|. On the other hand, if all ajµo(n)|Ajµo(n)| = |A| we get a contradiction to
the regularity of |A|. Therefore, (∗) ≺ |A|n.
Case II: Assume µo ∈ Fn \ Sn, then there exist at least two indices i1 6= i2 for which µo(i1) =
µo(i2) = jo. We show the existence of a permutation µl ∈ Sn that reaches the same ν-value for
Formula (3.8) as µo reaches, the proof is then completed by Case I, applied to µl.
13
Page 14
For the two components ai1jo |Aσo(i1)jo | and ai2jo |Aσo(i2)jo | of (3.8), indexed by µo(i1) = µo(i2) =
jo, we have the following combinatorial layouts:
jo jh
i1 ∗
σo(i1)
jo jh
σo(i2)
i2 ∗
The diagrams are useful to understand the modification of µo.
Since µ ∈ Mn \ Sn, there exists at least one index jh 6= jo in N \ Im(µo). Therefore, the
corresponding component, aijh |Aσo(i)jh |, is absent in (3.8). Without loss of generality, we take
ai2jo |Aσo(i2)jo | and modify it. Clearly, |Aσo(i2)jo | involves an entry a•jh , let ih be the index for
which σ(ih) = jh. Then |Aσo(i2)jo | = aihjh |Aσo(i2)ih,jojh |, and hence, by the maximality of µo,
ai2jo |Aσo(i2)jo | = ai2joaihjh |Aσo(i2)ih,jojh | = aihjh |Aihjh | .
Namely, we have specified another map µ1 ∈ Mn with µ1(i1) = jo, µ1(i2) = jh, and µ1(i) = µo(i)
for all i 6= i1, i2. Therefore, we reduced the number of indices sharing a same image in µo to have
Im(µo) ⊂ Im(µ1) ⊆ N . Proceeding inductively we get a chain
Im(µo) ⊂ Im(µ1) ⊂ · · · ⊂ Im(µl) = N,
the left equality is due to the finiteness of Fn. Thus µl ∈ Sn; the proof of Case II is then completed
by Case I.
So, we have showed that the identity σo is the single permutation that maximizes (3.6), and for
which we have B = AA▽ = |A|I˜. Since |A| ∈ R and I˜ is regular, so is B. This completes the proof
of Theorem 3.7 on pseudo invertibility of matrices over (T,⊕,⊙).
We push the result of Theorem 3.7 further:
Theorem 3.8. For each regular matrix A ∈ Mn(T), the products AA▽ and A▽A are idempotents.
Proof. Writing I˜ = AA▽, with I˜ = (ιij), we prove that I˜ = I˜2. Recall that a▽ij =
|Aji|
|A| , then
(3.9) ιij =
⊕
k
aika▽kj =
⊕
k
aik
|Ajk|
|A| = aike
|Ajke |
|A| ,
for some fixed ke. Suppose (I˜)2 = (ι(2)ij ), then
ι(2)ij =
⊕
h
ιihιhj =
⊕
h
(
⊕
k
aik
|Ahk|
|A|
)(
⊕
l
ahl
|Ajl|
|A|
)
=
(3.10)
⊕
h
⊕
k
⊕
l
aik
|Ahk|
|A| ahl
|Ajl|
|A| ,
and we need to prove the equality
(3.11) |A|
⊕
k
aik|Ajk| =
⊕
h
⊕
k
⊕
l
aik|Ahk|︸ ︷︷ ︸
(I)
ahl|Ajl|︸ ︷︷ ︸
(II)
.
To see that ι(2)ij ιij , take h = j to have (II) =
⊕
l ahl|Ajl| =
⊕
l ajl|Ajl| = |A|. By the way
of contradiction, assume ι(2)ij ≻ ιij , and suppose ko, ho, and lo are the indices reaching the ν-value
of ι(2)ij in Formula (3.10), then
aiko |Ahoko |aholo |Ajlo | ≻ aike |Ajke ||A| .
14
jo, we have the following combinatorial layouts:
jo jh
i1 ∗
σo(i1)
jo jh
σo(i2)
i2 ∗
The diagrams are useful to understand the modification of µo.
Since µ ∈ Mn \ Sn, there exists at least one index jh 6= jo in N \ Im(µo). Therefore, the
corresponding component, aijh |Aσo(i)jh |, is absent in (3.8). Without loss of generality, we take
ai2jo |Aσo(i2)jo | and modify it. Clearly, |Aσo(i2)jo | involves an entry a•jh , let ih be the index for
which σ(ih) = jh. Then |Aσo(i2)jo | = aihjh |Aσo(i2)ih,jojh |, and hence, by the maximality of µo,
ai2jo |Aσo(i2)jo | = ai2joaihjh |Aσo(i2)ih,jojh | = aihjh |Aihjh | .
Namely, we have specified another map µ1 ∈ Mn with µ1(i1) = jo, µ1(i2) = jh, and µ1(i) = µo(i)
for all i 6= i1, i2. Therefore, we reduced the number of indices sharing a same image in µo to have
Im(µo) ⊂ Im(µ1) ⊆ N . Proceeding inductively we get a chain
Im(µo) ⊂ Im(µ1) ⊂ · · · ⊂ Im(µl) = N,
the left equality is due to the finiteness of Fn. Thus µl ∈ Sn; the proof of Case II is then completed
by Case I.
So, we have showed that the identity σo is the single permutation that maximizes (3.6), and for
which we have B = AA▽ = |A|I˜. Since |A| ∈ R and I˜ is regular, so is B. This completes the proof
of Theorem 3.7 on pseudo invertibility of matrices over (T,⊕,⊙).
We push the result of Theorem 3.7 further:
Theorem 3.8. For each regular matrix A ∈ Mn(T), the products AA▽ and A▽A are idempotents.
Proof. Writing I˜ = AA▽, with I˜ = (ιij), we prove that I˜ = I˜2. Recall that a▽ij =
|Aji|
|A| , then
(3.9) ιij =
⊕
k
aika▽kj =
⊕
k
aik
|Ajk|
|A| = aike
|Ajke |
|A| ,
for some fixed ke. Suppose (I˜)2 = (ι(2)ij ), then
ι(2)ij =
⊕
h
ιihιhj =
⊕
h
(
⊕
k
aik
|Ahk|
|A|
)(
⊕
l
ahl
|Ajl|
|A|
)
=
(3.10)
⊕
h
⊕
k
⊕
l
aik
|Ahk|
|A| ahl
|Ajl|
|A| ,
and we need to prove the equality
(3.11) |A|
⊕
k
aik|Ajk| =
⊕
h
⊕
k
⊕
l
aik|Ahk|︸ ︷︷ ︸
(I)
ahl|Ajl|︸ ︷︷ ︸
(II)
.
To see that ι(2)ij ιij , take h = j to have (II) =
⊕
l ahl|Ajl| =
⊕
l ajl|Ajl| = |A|. By the way
of contradiction, assume ι(2)ij ≻ ιij , and suppose ko, ho, and lo are the indices reaching the ν-value
of ι(2)ij in Formula (3.10), then
aiko |Ahoko |aholo |Ajlo | ≻ aike |Ajke ||A| .
14
Page 15
Clearly aike |Ajke | aiko |Ajko |, since otherwise we would have a contradiction to the maximality
of (3.9). Thus,
aiko |Ahoko |aholo |Ajlo | ≻ aiko |Ajko ||A| ,
and hence
(3.12) |Ahoko |aholo |Ajlo | ≻ |Ajko ||A| .
Due to the maximality of |A|, we also have |A| ahoko |Ahoko |, and by (3.12) we get
|Ahoko |aholo |Ajlo | ≻ |Ajko ||A| |Ajko |ahoko |Ahoko | .
Namely,
(3.13) aholo |Ajlo | ≻ ahoko |Ajko | .
This contradicts the specification of ko as the index that reaches the maximum for (3.11). This
completes the proof that (AA▽)2 = AA▽; the case of multiplication on left is proved in the same
way.
Corollary 3.9. A matrix A 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
A =
(
1 −1
2 2
)
, then A▽ =
(
2 −1
2 1
)
(−3) ,
where |A| = 3. (Recall that, in tropical sense, multiplying by (−3) means dividing by 3.) The
product AA▽ is then
(
1 −1
2 2
)(
2 −1
2 1
)
(−3) =
(
3 0ν
4ν 3
)
(−3) =
(
0 (−3)ν
1ν 0
)
∈ U idmn (T) .
On the other hand, if we take the singular matrix
A =
(
1 −1
4 2
)
, then A▽ =
(
2 −1
4 1
)
(−3)ν ,
where here |A| = 3ν . Computing the product AA▽ we get
(
1 −1
4 2
)(
2 −1
4 1
)
(−3)ν =
(
3ν 0ν
6ν 3ν
)
(−3)ν /∈ Un(T) ,
which is not a regular matrix, and therefore AA▽ /∈ Un(T).
A few immediate conclusions are derived from our last results:
Corollary 3.11. Assume A is a regular matrix, then
(1) Adj(A) is also regular;
(2) |A| = (|A▽|)−1, and if A = A▽ then |A| = |A▽| = 0.
Proof. The first assertion is obvious. A, A▽, and AA▽ are all regular, then by Theorem 2.6
|A||A▽| = |I˜| = 0 and hence |A| = |A▽|−1.
The converse assertion of (2) is not true; for example, take the matrix
A =
(
−1 −2
−2 1
)
, then A▽ =
(
1 −2
−2 −1
)
.
Although |A| = |A▽| = 0, we have A 6= A▽.
15
of (3.9). Thus,
aiko |Ahoko |aholo |Ajlo | ≻ aiko |Ajko ||A| ,
and hence
(3.12) |Ahoko |aholo |Ajlo | ≻ |Ajko ||A| .
Due to the maximality of |A|, we also have |A| ahoko |Ahoko |, and by (3.12) we get
|Ahoko |aholo |Ajlo | ≻ |Ajko ||A| |Ajko |ahoko |Ahoko | .
Namely,
(3.13) aholo |Ajlo | ≻ ahoko |Ajko | .
This contradicts the specification of ko as the index that reaches the maximum for (3.11). This
completes the proof that (AA▽)2 = AA▽; the case of multiplication on left is proved in the same
way.
Corollary 3.9. A matrix A 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
A =
(
1 −1
2 2
)
, then A▽ =
(
2 −1
2 1
)
(−3) ,
where |A| = 3. (Recall that, in tropical sense, multiplying by (−3) means dividing by 3.) The
product AA▽ is then
(
1 −1
2 2
)(
2 −1
2 1
)
(−3) =
(
3 0ν
4ν 3
)
(−3) =
(
0 (−3)ν
1ν 0
)
∈ U idmn (T) .
On the other hand, if we take the singular matrix
A =
(
1 −1
4 2
)
, then A▽ =
(
2 −1
4 1
)
(−3)ν ,
where here |A| = 3ν . Computing the product AA▽ we get
(
1 −1
4 2
)(
2 −1
4 1
)
(−3)ν =
(
3ν 0ν
6ν 3ν
)
(−3)ν /∈ Un(T) ,
which is not a regular matrix, and therefore AA▽ /∈ Un(T).
A few immediate conclusions are derived from our last results:
Corollary 3.11. Assume A is a regular matrix, then
(1) Adj(A) is also regular;
(2) |A| = (|A▽|)−1, and if A = A▽ then |A| = |A▽| = 0.
Proof. The first assertion is obvious. A, A▽, and AA▽ are all regular, then by Theorem 2.6
|A||A▽| = |I˜| = 0 and hence |A| = |A▽|−1.
The converse assertion of (2) is not true; for example, take the matrix
A =
(
−1 −2
−2 1
)
, then A▽ =
(
1 −2
−2 −1
)
.
Although |A| = |A▽| = 0, we have A 6= A▽.
15
Page 16
Remark 3.12. Contrary to the classical theory of matrices over fields, tropically, the relation
(AB)▽ = B▽A▽ does not hold true; for example, take the regular matrix as in (2.4), then
A =
(
1 1
2 3
)
, A▽ =
(
3 1
2 1
)
(−4), and A▽A▽ =
(
6 4
5 3
)
(−8).
On the other hand, A2 is not regular, cf. Remark 2.4, and the computation of Adj(A2)/|A2| yields
(AA)▽ =
(
6 4
5 3
)
(−9)ν , where A2 =
(
3 4
5 6
)
;
this shows that (A▽)2 6= (AA)▽.
3.3. Matrices with real entries. Denoting by Mn(R¯) the semiring of matrices over (R¯,max,+ ),
the epimorphism pi : (T,⊕,⊙) −→ (R¯,max,+ ), cf. (1.9), induces in the standard way the epimor-
phism
pi∗ : Mn(T) −→ Mn(R¯)
of matrix semirings. We write pi∗(A) for the image of A ∈ Mn(T) in Mn(R¯). Conversely, set-
theoretic, Mn(R¯) ⊂ Mn(T).
Proposition 3.13. Suppose A ∈ Mn(T) is regular, where both A and A▽ have only real entries,
AA▽ = I˜ ′, and A▽A = I˜ ′′. Then
pi∗(I˜ ′A) = A, pi∗(A▽I˜ ′) = A▽, pi∗(I˜ ′′A▽) = A▽, and pi∗(AI˜ ′′) = A.
Proof. We prove the relation pi∗(I˜ ′A) = A. Letting I˜ ′ = (ιij), we show that
pi(
⊕
k
ιikakj) = aij ,
for all i, j. Recall that ιik = (
⊕
h aih|Ajh|) |A|−1, cf. Formula (3.5), and ιik ∈ R¯ν whenever i 6= k.
Composing together, we get
(*)
⊕
k
(
⊕
h
aih|Ajh||A|−1
)
akj =
⊕
k,h
aih|Ajh|akj
|A|−1 .
Using Formulas (3.7) and (3.8), we see that the maximal value of |Ajh|akj is attained when k =
h = j and it is |A|. Thus,
pi ((∗)) = pi(aij |Ajj |ajj |A|−1) = aij .
The other relations are proved in the same way.
Remark 3.14. In the sense of Proposition 3.15, the matrices I˜ ′ and I˜ ′′ are pseudo right/left
identities of A and A▽ respectively.
Pushing the results of Proposition 3.15 forward, we conclude:
Corollary 3.15. Suppose A ∈ Mn(T) is regular. Let AA▽ = I˜ ′ and A▽A = I˜ ′′; then
pi∗(I˜ ′A) = pi∗(A), pi∗(A▽I˜ ′) = pi∗(A▽), pi∗(I˜ ′′A▽) = pi∗(A▽), and pi∗(AI˜ ′′) = pi∗(A).
Example 3.16. Let
A =
(
1 1
2 3
)
then A▽ =
(
−1 −3
−2 −3
)
and AA▽ = I˜ ′ =
(
0 (−2)ν
1ν 0
)
.
Computing the products we have
I˜ ′A =
(
0 (−2)ν
1ν 0
)(
1 1
2 3
)
=
(
1 1ν
2ν 3
)
,
A▽I˜ ′ =
(
−1 −3
−2 −3
)(
0 (−2)ν
1ν 0
)
=
(
−1 (−3)ν
(−2)ν −3
)
,
and it easily verify that pi∗(I˜ ′A) = A and pi∗(A▽I˜ ′) = A▽.
16
(AB)▽ = B▽A▽ does not hold true; for example, take the regular matrix as in (2.4), then
A =
(
1 1
2 3
)
, A▽ =
(
3 1
2 1
)
(−4), and A▽A▽ =
(
6 4
5 3
)
(−8).
On the other hand, A2 is not regular, cf. Remark 2.4, and the computation of Adj(A2)/|A2| yields
(AA)▽ =
(
6 4
5 3
)
(−9)ν , where A2 =
(
3 4
5 6
)
;
this shows that (A▽)2 6= (AA)▽.
3.3. Matrices with real entries. Denoting by Mn(R¯) the semiring of matrices over (R¯,max,+ ),
the epimorphism pi : (T,⊕,⊙) −→ (R¯,max,+ ), cf. (1.9), induces in the standard way the epimor-
phism
pi∗ : Mn(T) −→ Mn(R¯)
of matrix semirings. We write pi∗(A) for the image of A ∈ Mn(T) in Mn(R¯). Conversely, set-
theoretic, Mn(R¯) ⊂ Mn(T).
Proposition 3.13. Suppose A ∈ Mn(T) is regular, where both A and A▽ have only real entries,
AA▽ = I˜ ′, and A▽A = I˜ ′′. Then
pi∗(I˜ ′A) = A, pi∗(A▽I˜ ′) = A▽, pi∗(I˜ ′′A▽) = A▽, and pi∗(AI˜ ′′) = A.
Proof. We prove the relation pi∗(I˜ ′A) = A. Letting I˜ ′ = (ιij), we show that
pi(
⊕
k
ιikakj) = aij ,
for all i, j. Recall that ιik = (
⊕
h aih|Ajh|) |A|−1, cf. Formula (3.5), and ιik ∈ R¯ν whenever i 6= k.
Composing together, we get
(*)
⊕
k
(
⊕
h
aih|Ajh||A|−1
)
akj =
⊕
k,h
aih|Ajh|akj
|A|−1 .
Using Formulas (3.7) and (3.8), we see that the maximal value of |Ajh|akj is attained when k =
h = j and it is |A|. Thus,
pi ((∗)) = pi(aij |Ajj |ajj |A|−1) = aij .
The other relations are proved in the same way.
Remark 3.14. In the sense of Proposition 3.15, the matrices I˜ ′ and I˜ ′′ are pseudo right/left
identities of A and A▽ respectively.
Pushing the results of Proposition 3.15 forward, we conclude:
Corollary 3.15. Suppose A ∈ Mn(T) is regular. Let AA▽ = I˜ ′ and A▽A = I˜ ′′; then
pi∗(I˜ ′A) = pi∗(A), pi∗(A▽I˜ ′) = pi∗(A▽), pi∗(I˜ ′′A▽) = pi∗(A▽), and pi∗(AI˜ ′′) = pi∗(A).
Example 3.16. Let
A =
(
1 1
2 3
)
then A▽ =
(
−1 −3
−2 −3
)
and AA▽ = I˜ ′ =
(
0 (−2)ν
1ν 0
)
.
Computing the products we have
I˜ ′A =
(
0 (−2)ν
1ν 0
)(
1 1
2 3
)
=
(
1 1ν
2ν 3
)
,
A▽I˜ ′ =
(
−1 −3
−2 −3
)(
0 (−2)ν
1ν 0
)
=
(
−1 (−3)ν
(−2)ν −3
)
,
and it easily verify that pi∗(I˜ ′A) = A and pi∗(A▽I˜ ′) = A▽.
16
Page 17
References
[1] P. Butkovic. Max-algebra: the linear algebra of combinatorics? Lin.Alg. and Appl., pages 313–335, 2003.
[2] R. Cuninghame-Green and P. Butkovic. Bases in max-algebra. Linear Algebra and its Applications, 389:107–
120, September 2004.
[3] M. Develin, F. Santos, and B. Sturmfels. On the rank of a tropical matrix. Preprint at arXiv:math.CO/0312114,
2004.
[4] A. Gathmann. Tropical algebraic geometry. Preprint at arXiv:math.AG/0601322, January 2006.
[5] J. Golan. The theory of semirings with applications in mathematics and theoretical computer science, volume 54.
Longman Sci & Tech., 1992.
[6] I. Itenberg, G. Mikhalkin, and E. Shustin. Tropical algebraic geometry, volume 35. Birkhauser, 2007. Oberwol-
fach seminars.
[7] K. H. Kim and F. W. Roush. Kapranov rank vs. tropical rank. Preprint at arXiv:math.CO/0503044, March
2005.
[8] G. Lallement. Semigroups and Combinatorial Applications. John Wiley & Sons, Inc., New York, NY, USA,
1979.
[9] E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, 1976.
[10] S. W. Margolis and J.-E. Pin. Inverse semigroups and extensions of groups by semilattices. J. Algebra, 110:277–
297, 1987.
[11] G. Mikhalkin. Atropical geometry and its applications. Proceedings of the ICM, Madrid, Spain, August 22-30,
2006 /Sanz-Sole´, Marta (ed.) et al., pages 827–852, 2006. Volume II: Invited lectures. Zu¨rich, European Math.
Soc.
[12] J. Richter-Gebert, B. Sturmfels, and T. Theobald. First steps in tropical geometry. Idempotent mathematics
and mathematical physics, pages 289–317, 2005. Contemp. Math., Amer. Math. Soc., Providence, RI, 377.
[13] E. Shustin. A tropical approach to enumerative geometry. Algebra i Analiz, 17(2):170–214, 2005.
[14] D. Speyer and B. Sturmfels. Tropical Grassmannians. Adv. Geom., 4(3):389–411, 2004.
[15] D. E. Speyer and B. Sturmfels. Tropical mathematics. Preprint at arXiv:math.CO/0408099, 2004.
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
CNRS et Universite Denis Diderot (Paris 7), 175, rue du Chevaleret 75013 Paris, France
E-mail address: zzur@math.biu.ac.il, zzur@post.tau.ac.il
17
[1] P. Butkovic. Max-algebra: the linear algebra of combinatorics? Lin.Alg. and Appl., pages 313–335, 2003.
[2] R. Cuninghame-Green and P. Butkovic. Bases in max-algebra. Linear Algebra and its Applications, 389:107–
120, September 2004.
[3] M. Develin, F. Santos, and B. Sturmfels. On the rank of a tropical matrix. Preprint at arXiv:math.CO/0312114,
2004.
[4] A. Gathmann. Tropical algebraic geometry. Preprint at arXiv:math.AG/0601322, January 2006.
[5] J. Golan. The theory of semirings with applications in mathematics and theoretical computer science, volume 54.
Longman Sci & Tech., 1992.
[6] I. Itenberg, G. Mikhalkin, and E. Shustin. Tropical algebraic geometry, volume 35. Birkhauser, 2007. Oberwol-
fach seminars.
[7] K. H. Kim and F. W. Roush. Kapranov rank vs. tropical rank. Preprint at arXiv:math.CO/0503044, March
2005.
[8] G. Lallement. Semigroups and Combinatorial Applications. John Wiley & Sons, Inc., New York, NY, USA,
1979.
[9] E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, 1976.
[10] S. W. Margolis and J.-E. Pin. Inverse semigroups and extensions of groups by semilattices. J. Algebra, 110:277–
297, 1987.
[11] G. Mikhalkin. Atropical geometry and its applications. Proceedings of the ICM, Madrid, Spain, August 22-30,
2006 /Sanz-Sole´, Marta (ed.) et al., pages 827–852, 2006. Volume II: Invited lectures. Zu¨rich, European Math.
Soc.
[12] J. Richter-Gebert, B. Sturmfels, and T. Theobald. First steps in tropical geometry. Idempotent mathematics
and mathematical physics, pages 289–317, 2005. Contemp. Math., Amer. Math. Soc., Providence, RI, 377.
[13] E. Shustin. A tropical approach to enumerative geometry. Algebra i Analiz, 17(2):170–214, 2005.
[14] D. Speyer and B. Sturmfels. Tropical Grassmannians. Adv. Geom., 4(3):389–411, 2004.
[15] D. E. Speyer and B. Sturmfels. Tropical mathematics. Preprint at arXiv:math.CO/0408099, 2004.
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
CNRS et Universite Denis Diderot (Paris 7), 175, rue du Chevaleret 75013 Paris, France
E-mail address: zzur@math.biu.ac.il, zzur@post.tau.ac.il
17
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