Semialgebra systems: An algebraic structure theory applicable to tropical mathematics Shirshov 100 Conference.

Louis Rowen, Bar-Ilan University

16 August, 2021 Ode to Shirshov

I first heard of Shirshov from Amitsur (also born the same year!) who discovered the amazing Shirshov Height Theorem, which revolutionized PI-theory. I did not have the opportunity to meet Shirshov, but have the privilege of knowing several of his students, including Bokut (who has been a frequent guest at Bar-Ilan University and introduced me to many members of the important algebra group in Novosibirsk), Zelmanov, Kemer, (both joint students of Bokut and Shirshov), and Shestakov. The main original idea was to take the limit of the logarithm of the absolute values of the coordinates of an affine variety as the base of the logarithm goes to ∞.

Tropical background in 3 slides

Tropical geometry has assumed a prominent position in mathematics because of its ability to simplify algebraic geometry while not changing some invariants (involving intersection numbers of varieties), thereby simplifying difficult computations. Tropical background in 3 slides

Tropical geometry has assumed a prominent position in mathematics because of its ability to simplify algebraic geometry while not changing some invariants (involving intersection numbers of varieties), thereby simplifying difficult computations. The main original idea was to take the limit of the logarithm of the absolute values of the coordinates of an affine variety as the base of the logarithm goes to ∞. Tropicalization

For any complex affine variety W = {(z1,..., zn): zi ∈ C}, and any t > 0, its amoeba A(W ) is defined as

{(logt |z1|,..., logt |zn|) :(z1,..., zn) ∈ W } (n) ⊂ (R ∪ {−∞}) , graphed according to the (rescaled) coordinates logt |z1|,..., logt |zn|. Thus, respectively, multiplication has been replaced by addition, whereas addition has been replaced by the maximum. The degeneration t → ∞ is called tropicalization, honoring Imre Simon, who worked in Brazil.

a b a b Note that logt (t t ) = logt (t ) + logt (t ).

Also, for a = b + c

a b b c c logt (t + t ) = logt (t (t + 1)) = b + logt (t + 1)

a ≈ b + c = a = logt (t ), as t → ∞. a b a b Note that logt (t t ) = logt (t ) + logt (t ).

Also, for a = b + c

a b b c c logt (t + t ) = logt (t (t + 1)) = b + logt (t + 1)

a ≈ b + c = a = logt (t ), as t → ∞. Thus, respectively, multiplication has been replaced by addition, whereas addition has been replaced by the maximum. The degeneration t → ∞ is called tropicalization, honoring Imre Simon, who worked in Brazil. Any ordered group (G, ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation.

For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5.

We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 .

Semirings

The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be max{a, b}. This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A\{0}, ·) is a group. For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5.

We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 .

Semirings

The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be max{a, b}. This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A\{0}, ·) is a group.

Any ordered group (G, ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation. We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 .

Semirings

The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be max{a, b}. This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A\{0}, ·) is a group.

Any ordered group (G, ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation.

For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5. Semirings

The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be max{a, b}. This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A\{0}, ·) is a group.

Any ordered group (G, ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation.

For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5.

We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 . For any set S, Fun(S, A) is the semiring of functions from S to A, with operations defined elementwise. Likewise one can define the Weyl semialgebra over a commutative semiring A, by the relation yx = xy + 1. One also can define modules over semirings in the usual way, sometimes called “semimodules.” The free A-module of rank n is just the Cartesian product A(n), with operations componentwise.

Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). Likewise one can define the Weyl semialgebra over a commutative semiring A, by the relation yx = xy + 1. One also can define modules over semirings in the usual way, sometimes called “semimodules.” The free A-module of rank n is just the Cartesian product A(n), with operations componentwise.

Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). For any set S, Fun(S, A) is the semiring of functions from S to A, with operations defined elementwise. One also can define modules over semirings in the usual way, sometimes called “semimodules.” The free A-module of rank n is just the Cartesian product A(n), with operations componentwise.

Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). For any set S, Fun(S, A) is the semiring of functions from S to A, with operations defined elementwise. Likewise one can define the Weyl semialgebra over a commutative semiring A, by the relation yx = xy + 1. Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). For any set S, Fun(S, A) is the semiring of functions from S to A, with operations defined elementwise. Likewise one can define the Weyl semialgebra over a commutative semiring A, by the relation yx = xy + 1. One also can define modules over semirings in the usual way, sometimes called “semimodules.” The free A-module of rank n is just the Cartesian product A(n), with operations componentwise. The characteristic char(A) of A is the minimal n such that k + n = k for some k; if there is none such, then char(A) = 0.

Nmax has characteristic 1 since1+1=1 . (But in this definition one could have1+1 6=1 with2+1=2 .) Characteristic 1 geometry has been quite popular in recent years.

Every semiring A has a sub-semiring additively generated by1. We write n := n1. Note that there is always a semiring homomorphism N → A given by n 7→ n for each n. Nmax has characteristic 1 since1+1=1 . (But in this definition one could have1+1 6=1 with2+1=2 .) Characteristic 1 geometry has been quite popular in recent years.

Every semiring A has a sub-semiring additively generated by1. We write n := n1. Note that there is always a semiring homomorphism N → A given by n 7→ n for each n.

The characteristic char(A) of A is the minimal n such that k + n = k for some k; if there is none such, then char(A) = 0. Every semiring A has a sub-semiring additively generated by1. We write n := n1. Note that there is always a semiring homomorphism N → A given by n 7→ n for each n.

The characteristic char(A) of A is the minimal n such that k + n = k for some k; if there is none such, then char(A) = 0.

Nmax has characteristic 1 since1+1=1 . (But in this definition one could have1+1 6=1 with2+1=2 .) Characteristic 1 geometry has been quite popular in recent years. It follows that one has an infinite chain of subspaces all of dimension 2, and one must take this into account in the theory of vector spaces.

Vector spaces over Rmax

A vector space generated by a single point p is just a line Lp at an angle of 45◦ through p. The space generated by p and q is then the strip between Lp and Lq. Vector spaces over Rmax

A vector space generated by a single point p is just a line Lp at an angle of 45◦ through p. The space generated by p and q is then the strip between Lp and Lq. It follows that one has an infinite chain of subspaces all of dimension 2, and one must take this into account in the theory of vector spaces. Some drawbacks of the max-plus algebra

1. Almost since the origins of tropical algebra, it was understood that the max-plus algebra does not work so well, since one cannot take logarithms in C. There are ingenious ways around this, such as patchworking, but they are rather complicated. Some drawbacks of the max-plus algebra

1. Almost since the origins of tropical algebra, it was understood that the max-plus algebra does not work so well, since one cannot take logarithms in C. There are ingenious ways around this, such as patchworking, but they are rather complicated. but

0 · 0 + 0 · 1 0 · 0 + 0 · 2 1 2 A2 = = 1 · 0 + 2 · 1 1 · 0 + 2 · 2 3 4

has perminant 5 > 22.

2. Without negation, the naive approach to determinants of matrices would be to use the permanent (defined without minus) but then one must contend with matrices like 0 0 A = , (1) 1 2 where A has permanent 2 2. Without negation, the naive approach to determinants of matrices would be to use the permanent (defined without minus) but then one must contend with matrices like 0 0 A = , (1) 1 2 where A has permanent 2 but

0 · 0 + 0 · 1 0 · 0 + 0 · 2 1 2 A2 = = 1 · 0 + 2 · 1 1 · 0 + 2 · 2 3 4 has perminant 5 > 22. 3. One can never reach0 in a nontrivial algebraic expression, which wreaks havoc on algebraicity and roots. (A combinatoric way out has been to define “corner roots,” but this leads to difficulty in understanding basic algebraic concepts) C{{t}} is algebraically closed, so is “elementarily equivalent” to C in the language of logic.

Puiseux series and tropicalization

Over the years, a more successful model came about. For a field K, the field K = K{{t}} of Puiseux series on the variable t, is the set P∞ j/N of formal series of the form f = j=` cj t where N ∈ N, ` ∈ Z, and cj ∈ S (with the standard convolution product). Then one has the Puiseux valuation val : K{{t}} \ {0} → Q ⊂ R defined by

val(f ) = − min{k/N}, (2) cj 6=0

which we also call tropicalization. Puiseux series and tropicalization

Over the years, a more successful model came about. For a field K, the field K = K{{t}} of Puiseux series on the variable t, is the set P∞ j/N of formal series of the form f = j=` cj t where N ∈ N, ` ∈ Z, and cj ∈ S (with the standard convolution product). Then one has the Puiseux valuation val : K{{t}} \ {0} → Q ⊂ R defined by

val(f ) = − min{k/N}, (2) cj 6=0

which we also call tropicalization.

C{{t}} is algebraically closed, so is “elementarily equivalent” to C in the language of logic. For example, if f = 2λ2 + 7λ4 and g = −2λ2 + 5λ3 + 7λ4 then f + g = 5λ3 + 14λ4 and − val(f + g) = 3 > 2. Thus, the negatives of valuations behave like the max-plus algebra EXCEPT perhaps when evaluated on elements having the same value.

Customarily the target Q of − val has been identified with the max-plus algebra, Zmax, but this is inaccurate. Although −(val(f ) + val(g)) = max{− val(f ), − val(g)} when val(f ) 6= val(g), this can fail when val(f ) = val(g), due to cancelation in the lowest terms of f and g. Thus, the negatives of valuations behave like the max-plus algebra EXCEPT perhaps when evaluated on elements having the same value.

Customarily the target Q of − val has been identified with the max-plus algebra, Zmax, but this is inaccurate. Although −(val(f ) + val(g)) = max{− val(f ), − val(g)} when val(f ) 6= val(g), this can fail when val(f ) = val(g), due to cancelation in the lowest terms of f and g. For example, if f = 2λ2 + 7λ4 and g = −2λ2 + 5λ3 + 7λ4 then f + g = 5λ3 + 14λ4 and − val(f + g) = 3 > 2. Customarily the target Q of − val has been identified with the max-plus algebra, Zmax, but this is inaccurate. Although −(val(f ) + val(g)) = max{− val(f ), − val(g)} when val(f ) 6= val(g), this can fail when val(f ) = val(g), due to cancelation in the lowest terms of f and g. For example, if f = 2λ2 + 7λ4 and g = −2λ2 + 5λ3 + 7λ4 then f + g = 5λ3 + 14λ4 and − val(f + g) = 3 > 2. Thus, the negatives of valuations behave like the max-plus algebra EXCEPT perhaps when evaluated on elements having the same value. Supertropical algebra

To remedy the failings of the max-plus, Izhakian (anticipated by Dress) brought in supertropical algebra. We start with a multiplicative monoid T , which we want to study, and embed it in a larger semiring including a copy G of T . The standard supertropical semifield is a quadruple (A, T , G =∼ T , ν) where G ⊂ A is an ordered submonoid, G ⊂ A is a semiring ideal, and A := T ∪ G, with a projection ν : A → G, satisfying ν2 = ν and restricting to an isomorphism T → G as well as the condition, writing a◦ for ν(a):  a whenever a◦ > b◦,  a + b = b whenever a◦ < b◦, a◦ whenever a◦ = b◦. Summarizing, we start with T , which has some partial algebraic structure but not enough for our purposes, and pass to the standard supertropical semifield A by adjoining a “ghost” copy of T . This is our main model for the tropical theory. Note that the standard supertropical semifield A has characteristic 1, since 1◦ +1=1 ◦. A crucial new feature is the ghost surpassing map , where b  a if b = a + c◦ for some c, which means either b = a, or 1 2 b = c◦ for c > a. In our previous example now has 3 4 permanent 5 + 5 = 5◦, and 5◦  4. The “correct theorem, taking the tropical determinant to be the permanent, is :

det(AB)  det(A) det(B).

The elements of G are called ghost elements and ν : A → G is called the ghost map. T is the monoid of tangible elements, and encapsulates the tropical aspect. A crucial new feature is the ghost surpassing map , where b  a if b = a + c◦ for some c, which means either b = a, or 1 2 b = c◦ for c > a. In our previous example now has 3 4 permanent 5 + 5 = 5◦, and 5◦  4. The “correct theorem, taking the tropical determinant to be the permanent, is :

det(AB)  det(A) det(B).

The elements of G are called ghost elements and ν : A → G is called the ghost map. T is the monoid of tangible elements, and encapsulates the tropical aspect. Summarizing, we start with T , which has some partial algebraic structure but not enough for our purposes, and pass to the standard supertropical semifield A by adjoining a “ghost” copy of T . This is our main model for the tropical theory. Note that the standard supertropical semifield A has characteristic 1, since 1◦ +1=1 ◦. det(AB)  det(A) det(B).

The elements of G are called ghost elements and ν : A → G is called the ghost map. T is the monoid of tangible elements, and encapsulates the tropical aspect. Summarizing, we start with T , which has some partial algebraic structure but not enough for our purposes, and pass to the standard supertropical semifield A by adjoining a “ghost” copy of T . This is our main model for the tropical theory. Note that the standard supertropical semifield A has characteristic 1, since 1◦ +1=1 ◦. A crucial new feature is the ghost surpassing map , where b  a if b = a + c◦ for some c, which means either b = a, or 1 2 b = c◦ for c > a. In our previous example now has 3 4 permanent 5 + 5 = 5◦, and 5◦  4. The “correct theorem, taking the tropical determinant to be the permanent, is : The elements of G are called ghost elements and ν : A → G is called the ghost map. T is the monoid of tangible elements, and encapsulates the tropical aspect. Summarizing, we start with T , which has some partial algebraic structure but not enough for our purposes, and pass to the standard supertropical semifield A by adjoining a “ghost” copy of T . This is our main model for the tropical theory. Note that the standard supertropical semifield A has characteristic 1, since 1◦ +1=1 ◦. A crucial new feature is the ghost surpassing map , where b  a if b = a + c◦ for some c, which means either b = a, or 1 2 b = c◦ for c > a. In our previous example now has 3 4 permanent 5 + 5 = 5◦, and 5◦  4. The “correct theorem, taking the tropical determinant to be the permanent, is :

det(AB)  det(A) det(B). This motivated an ongoing project to present an axiomatic algebraic theory which unifies and “explains” these parallels, by starting with some monoid T that lacks enough algebraic structure to explore it fully, and embedding T in a semiring A endowed with more structure. T is called the set of tangible elements of A, our ultimate object. Usually A is a (not necessarily associative) semiring. So now we start again.

Izhakian and R found a supertropical Nullstellensatz. Next, we studied matrices (and, with Knebusch) linear algebra over the supertropical semifield, and found that many theorems from classical algebra have mysterious analogs for supertropical algebra. So now we start again.

Izhakian and R found a supertropical Nullstellensatz. Next, we studied matrices (and, with Knebusch) linear algebra over the supertropical semifield, and found that many theorems from classical algebra have mysterious analogs for supertropical algebra. This motivated an ongoing project to present an axiomatic algebraic theory which unifies and “explains” these parallels, by starting with some monoid T that lacks enough algebraic structure to explore it fully, and embedding T in a semiring A endowed with more structure. T is called the set of tangible elements of A, our ultimate object. Usually A is a (not necessarily associative) semiring. Izhakian and R found a supertropical Nullstellensatz. Next, we studied matrices (and, with Knebusch) linear algebra over the supertropical semifield, and found that many theorems from classical algebra have mysterious analogs for supertropical algebra. This motivated an ongoing project to present an axiomatic algebraic theory which unifies and “explains” these parallels, by starting with some monoid T that lacks enough algebraic structure to explore it fully, and embedding T in a semiring A endowed with more structure. T is called the set of tangible elements of A, our ultimate object. Usually A is a (not necessarily associative) semiring. So now we start again. 2. A is a graded algebra, such as a polynomial algebra or an exterior algebra, and T is the monoid of homogeneous elements. 3. A is a vector space with base T . More specifically, A may be an algebra with a multiplicative base T . For example, A could be the group algebra of a group T . 4. A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 5. A is the set of class functions from a finite group to a field F ; T0 is the sub-semiring of characters. (This can be generalized to association schemes.) 6. T is a hyperfield (to be defined presently), and A ⊆ P(T ), the power set (the set of subsets of T .)

Classical illustrations The following list of examples gives a range of relevant situations. In many examples but not all, T is a monoid. 1. (A fundamental classical example) A = T ∪ {0} is an integral domain. 3. A is a vector space with base T . More specifically, A may be an algebra with a multiplicative base T . For example, A could be the group algebra of a group T . 4. A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 5. A is the set of class functions from a finite group to a field F ; T0 is the sub-semiring of characters. (This can be generalized to association schemes.) 6. T is a hyperfield (to be defined presently), and A ⊆ P(T ), the power set (the set of subsets of T .)

Classical illustrations The following list of examples gives a range of relevant situations. In many examples but not all, T is a monoid. 1. (A fundamental classical example) A = T ∪ {0} is an integral domain. 2. A is a graded algebra, such as a polynomial algebra or an exterior algebra, and T is the monoid of homogeneous elements. 4. A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 5. A is the set of class functions from a finite group to a field F ; T0 is the sub-semiring of characters. (This can be generalized to association schemes.) 6. T is a hyperfield (to be defined presently), and A ⊆ P(T ), the power set (the set of subsets of T .)

Classical illustrations The following list of examples gives a range of relevant situations. In many examples but not all, T is a monoid. 1. (A fundamental classical example) A = T ∪ {0} is an integral domain. 2. A is a graded algebra, such as a polynomial algebra or an exterior algebra, and T is the monoid of homogeneous elements. 3. A is a vector space with base T . More specifically, A may be an algebra with a multiplicative base T . For example, A could be the group algebra of a group T . 5. A is the set of class functions from a finite group to a field F ; T0 is the sub-semiring of characters. (This can be generalized to association schemes.) 6. T is a hyperfield (to be defined presently), and A ⊆ P(T ), the power set (the set of subsets of T .)

Classical illustrations The following list of examples gives a range of relevant situations. In many examples but not all, T is a monoid. 1. (A fundamental classical example) A = T ∪ {0} is an integral domain. 2. A is a graded algebra, such as a polynomial algebra or an exterior algebra, and T is the monoid of homogeneous elements. 3. A is a vector space with base T . More specifically, A may be an algebra with a multiplicative base T . For example, A could be the group algebra of a group T . 4. A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 6. T is a hyperfield (to be defined presently), and A ⊆ P(T ), the power set (the set of subsets of T .)

Classical illustrations The following list of examples gives a range of relevant situations. In many examples but not all, T is a monoid. 1. (A fundamental classical example) A = T ∪ {0} is an integral domain. 2. A is a graded algebra, such as a polynomial algebra or an exterior algebra, and T is the monoid of homogeneous elements. 3. A is a vector space with base T . More specifically, A may be an algebra with a multiplicative base T . For example, A could be the group algebra of a group T . 4. A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 5. A is the set of class functions from a finite group to a field F ; T0 is the sub-semiring of characters. (This can be generalized to association schemes.) Classical illustrations The following list of examples gives a range of relevant situations. In many examples but not all, T is a monoid. 1. (A fundamental classical example) A = T ∪ {0} is an integral domain. 2. A is a graded algebra, such as a polynomial algebra or an exterior algebra, and T is the monoid of homogeneous elements. 3. A is a vector space with base T . More specifically, A may be an algebra with a multiplicative base T . For example, A could be the group algebra of a group T . 4. A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 5. A is the set of class functions from a finite group to a field F ; T0 is the sub-semiring of characters. (This can be generalized to association schemes.) 6. T is a hyperfield (to be defined presently), and A ⊆ P(T ), the power set (the set of subsets of T .) I What is the basic algebraic structure on which to pin the theory?

I How can we develop linear algebra to obtain analogs of the main theorems of classical matrix theory?

I What is an (affine, projective) variety in this framework? (We would start with an algebraic definition of “root” that matches geometric intuition.)

I Is there a version of module theory that could support a homological theory?

Our overall goal in this study is to provide an algebraic umbrella, which includes as much of the classical theory as possible, with the goal of addressing the basic questions such as the following: Our overall goal in this study is to provide an algebraic umbrella, which includes as much of the classical theory as possible, with the goal of addressing the basic questions such as the following:

I What is the basic algebraic structure on which to pin the theory?

I How can we develop linear algebra to obtain analogs of the main theorems of classical matrix theory?

I What is an (affine, projective) variety in this framework? (We would start with an algebraic definition of “root” that matches geometric intuition.)

I Is there a version of module theory that could support a homological theory? Definition: A surpassing relation on A, denoted , is a partial pre-order 0 0 satisfying the following, for elements a, a ∈ T , and b, b , bi ∈ A: 1.0  b◦. 0 0 0 2. If bi  bi for i = 1, 2 then b1 + b2  b1 + b2. 3. If a  a0 then a = a0. 4. b◦ 6 a.

It follows from (1) and (2) that b  b0 whenever b + c◦ = b0 for some c ∈ A◦.

The surpassing relation Since it does not help in equating expressions to0 , we obtain parallels of the classical algebraic theory by generalizing equality on T to a relation on A which is not symmetric, but nevertheless provides a robust algebraic theory. This key ingredient is a surpassing relation , to replace equality in our theorems, which now becomes one-sided, but replaces = in our theorems! It follows from (1) and (2) that b  b0 whenever b + c◦ = b0 for some c ∈ A◦.

The surpassing relation Since it does not help in equating expressions to0 , we obtain parallels of the classical algebraic theory by generalizing equality on T to a relation on A which is not symmetric, but nevertheless provides a robust algebraic theory. This key ingredient is a surpassing relation , to replace equality in our theorems, which now becomes one-sided, but replaces = in our theorems!

Definition: A surpassing relation on A, denoted , is a partial pre-order 0 0 satisfying the following, for elements a, a ∈ T , and b, b , bi ∈ A: 1.0  b◦. 0 0 0 2. If bi  bi for i = 1, 2 then b1 + b2  b1 + b2. 3. If a  a0 then a = a0. 4. b◦ 6 a. The surpassing relation Since it does not help in equating expressions to0 , we obtain parallels of the classical algebraic theory by generalizing equality on T to a relation on A which is not symmetric, but nevertheless provides a robust algebraic theory. This key ingredient is a surpassing relation , to replace equality in our theorems, which now becomes one-sided, but replaces = in our theorems!

Definition: A surpassing relation on A, denoted , is a partial pre-order 0 0 satisfying the following, for elements a, a ∈ T , and b, b , bi ∈ A: 1.0  b◦. 0 0 0 2. If bi  bi for i = 1, 2 then b1 + b2  b1 + b2. 3. If a  a0 then a = a0. 4. b◦ 6 a.

It follows from (1) and (2) that b  b0 whenever b + c◦ = b0 for some c ∈ A◦. Definition: The ◦- relation ◦ is the relation given by a ◦ b iff b = a + c for some c ∈ A◦.

One can check that ◦ is indeed a surpassing relation in any metatangible triple.

More generally, given an ideal I, define a I b iff b = a + c for some c ∈ I. (But we shall see other examples.) Thus I and A◦ play analogous roles to the ideal {0}. The next condition is close to max-plus, but differs in one crucial case. A is I -bipotent if a + b ∈ {a, b} whenever a, b ∈ A with a + b ∈/ I.

The last condition can be used to define the main example of a surpassing relation. More generally, given an ideal I, define a I b iff b = a + c for some c ∈ I. (But we shall see other examples.) Thus I and A◦ play analogous roles to the ideal {0}. The next condition is close to max-plus, but differs in one crucial case. A is I -bipotent if a + b ∈ {a, b} whenever a, b ∈ A with a + b ∈/ I.

The last condition can be used to define the main example of a surpassing relation. Definition: The ◦- relation ◦ is the relation given by a ◦ b iff b = a + c for some c ∈ A◦.

One can check that ◦ is indeed a surpassing relation in any metatangible triple. Thus I and A◦ play analogous roles to the ideal {0}. The next condition is close to max-plus, but differs in one crucial case. A is I -bipotent if a + b ∈ {a, b} whenever a, b ∈ A with a + b ∈/ I.

The last condition can be used to define the main example of a surpassing relation. Definition: The ◦- relation ◦ is the relation given by a ◦ b iff b = a + c for some c ∈ A◦.

One can check that ◦ is indeed a surpassing relation in any metatangible triple.

More generally, given an ideal I, define a I b iff b = a + c for some c ∈ I. (But we shall see other examples.) The next condition is close to max-plus, but differs in one crucial case. A is I -bipotent if a + b ∈ {a, b} whenever a, b ∈ A with a + b ∈/ I.

The last condition can be used to define the main example of a surpassing relation. Definition: The ◦- relation ◦ is the relation given by a ◦ b iff b = a + c for some c ∈ A◦.

One can check that ◦ is indeed a surpassing relation in any metatangible triple.

More generally, given an ideal I, define a I b iff b = a + c for some c ∈ I. (But we shall see other examples.) Thus I and A◦ play analogous roles to the ideal {0}. A is I -bipotent if a + b ∈ {a, b} whenever a, b ∈ A with a + b ∈/ I.

The last condition can be used to define the main example of a surpassing relation. Definition: The ◦- relation ◦ is the relation given by a ◦ b iff b = a + c for some c ∈ A◦.

One can check that ◦ is indeed a surpassing relation in any metatangible triple.

More generally, given an ideal I, define a I b iff b = a + c for some c ∈ I. (But we shall see other examples.) Thus I and A◦ play analogous roles to the ideal {0}. The next condition is close to max-plus, but differs in one crucial case. The last condition can be used to define the main example of a surpassing relation. Definition: The ◦- relation ◦ is the relation given by a ◦ b iff b = a + c for some c ∈ A◦.

One can check that ◦ is indeed a surpassing relation in any metatangible triple.

More generally, given an ideal I, define a I b iff b = a + c for some c ∈ I. (But we shall see other examples.) Thus I and A◦ play analogous roles to the ideal {0}. The next condition is close to max-plus, but differs in one crucial case. A is I -bipotent if a + b ∈ {a, b} whenever a, b ∈ A with a + b ∈/ I. Equality of row rank and column rank (and submatrix rank, defined as the largest size of a square submatrix A0 with (A0) ∈/ I) holds over the standard supertropical algebra, but not in many other cases to be discussed. The situation is studied in depth in joint work with Akian and Gaubert.

()-Linear algebra

Write V := A(n), where (A, T , ) are given. We define a set of P vectors {vi ∈ V : i ∈ I } to be T -dependent if i∈I 0 αi vi  (0) for 0 some nonempty subset I ⊆ I and αi ∈ T , and the row rank of a matrix to be the maximal number of T -independent rows. The notion of column rank is analogous. ()-Linear algebra

Write V := A(n), where (A, T , ) are given. We define a set of P vectors {vi ∈ V : i ∈ I } to be T -dependent if i∈I 0 αi vi  (0) for 0 some nonempty subset I ⊆ I and αi ∈ T , and the row rank of a matrix to be the maximal number of T -independent rows. The notion of column rank is analogous. Equality of row rank and column rank (and submatrix rank, defined as the largest size of a square submatrix A0 with (A0) ∈/ I) holds over the standard supertropical algebra, but not in many other cases to be discussed. The situation is studied in depth in joint work with Akian and Gaubert. ()-exterior algebras

The ()-exterior algebra over a free module V over a semifield is the algebra generated (by juxtaposition) by terms v1 ∧ · · · ∧ vn, where we declare v ∧ v  0 and v ∧ w + w ∧ v  0 for each v, w ∈ V . We write a(−)b for a + ((−)b).

Our last ingredient: Negation maps

Up to now negation has been notably absent, and although we can define semirings without it, we cannot get far without it, unless we impose very restrictive conditions. In order to overcome this obstacle, we introduce a formal negation map a 7→ (−)a, which satisfies all of the properties of negation except a + ((−)a) =0. To wit: A negation map (−) is an additive homomorphism (−):(A, +) → (A, +) of order ≤ 2, written a 7→ (−)a, satisfying

(−)(a1a2) = ((−)a1)a2 = a1((−)a2), ∀a ∈ A. Our last ingredient: Negation maps

Up to now negation has been notably absent, and although we can define semirings without it, we cannot get far without it, unless we impose very restrictive conditions. In order to overcome this obstacle, we introduce a formal negation map a 7→ (−)a, which satisfies all of the properties of negation except a + ((−)a) =0. To wit: A negation map (−) is an additive homomorphism (−):(A, +) → (A, +) of order ≤ 2, written a 7→ (−)a, satisfying

(−)(a1a2) = ((−)a1)a2 = a1((−)a2), ∀a ∈ A.

We write a(−)b for a + ((−)b). We require that a◦  0 for every a ∈ A. Quasi-zeroes play a crucial role, replacing0 in the theorems. In the classical theory, where (−) is the usual negation, the only quasi-zero is0 . BUT a perfectly good negation map is the identity map (used in supertropical!)

Given a ∈ A we define the quasi-zero a◦ := a(−)a, and

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a. In the classical theory, where (−) is the usual negation, the only quasi-zero is0 . BUT a perfectly good negation map is the identity map (used in supertropical!)

Given a ∈ A we define the quasi-zero a◦ := a(−)a, and

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a. We require that a◦  0 for every a ∈ A. Quasi-zeroes play a crucial role, replacing0 in the theorems. BUT a perfectly good negation map is the identity map (used in supertropical!)

Given a ∈ A we define the quasi-zero a◦ := a(−)a, and

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a. We require that a◦  0 for every a ∈ A. Quasi-zeroes play a crucial role, replacing0 in the theorems. In the classical theory, where (−) is the usual negation, the only quasi-zero is0 . Given a ∈ A we define the quasi-zero a◦ := a(−)a, and

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a. We require that a◦  0 for every a ∈ A. Quasi-zeroes play a crucial role, replacing0 in the theorems. In the classical theory, where (−) is the usual negation, the only quasi-zero is0 . BUT a perfectly good negation map is the identity map (used in supertropical!) One of the key axioms: A triple (A, T , (−)) has unique quasi-negatives if a(−)b ∈ A◦ for a, b ∈ T implies b = a.

Triples

(A, T , (−)) where (−) is a negation map with (−)T = T is called a triple, when: 1. T generates A additively, and 2. T ∩ A◦ = ∅. In particular0 ∈/ T . We write T0 for T ∪ {0}. Triples

(A, T , (−)) where (−) is a negation map with (−)T = T is called a triple, when: 1. T generates A additively, and 2. T ∩ A◦ = ∅. In particular0 ∈/ T . We write T0 for T ∪ {0}. One of the key axioms: A triple (A, T , (−)) has unique quasi-negatives if a(−)b ∈ A◦ for a, b ∈ T implies b = a. The identity itself is a perfectly valid negation map (since one just erases the minus signs in the definition), of first kind, which is used in supertropical algebra; then a◦ = ν(a).

The negation map (−) is said to be of the first kind if (−)a = a for all a ∈ T (and thus (−) is the identity), which occurs in the supertropical situation, and of the second kind if (−)a 6= a for all a ∈ T . The negation map (−) is said to be of the first kind if (−)a = a for all a ∈ T (and thus (−) is the identity), which occurs in the supertropical situation, and of the second kind if (−)a 6= a for all a ∈ T . The identity itself is a perfectly valid negation map (since one just erases the minus signs in the definition), of first kind, which is used in supertropical algebra; then a◦ = ν(a). Symmetrized triples

There is a very useful way of introducing negation maps of the second kind, when we do not have one, along the lines of the familiar Bourbaki-Grothendieck construction based on the classical way of constructing Z from N, by taking ordered pairs (m, n) and modding out the equivalence identifying (m1, n1) and (m2, n2) when m1 + n2 = m2 + n1. Here we exploit the same equivalence but do not mod out by it (since everything would degenerate). The symmetrization process, obtained by Gaubert (1992) in his dissertation, embeds any semiring into a semiring with a natural negation map of second kind, thereby providing a functor from semirings to triples. Define Tb = (T × {0}) ∪ ({0} × T ), a monoid when T is a monoid, and a group whenever T is. A embeds into Ab by means of the first component.

Given any T -module A, define Ab to be A × A, with componentwise addition. Also define “twist” multiplication (T × T ) × Ab → Ab given by

(a0, a1)(b0, b1) = (a0b0 + a1b1, a0b1 + a0b1). Given any T -module A, define Ab to be A × A, with componentwise addition. Also define “twist” multiplication (T × T ) × Ab → Ab given by

(a0, a1)(b0, b1) = (a0b0 + a1b1, a0b1 + a0b1).

Define Tb = (T × {0}) ∪ ({0} × T ), a monoid when T is a monoid, and a group whenever T is. A embeds into Ab by means of the first component. In particular, Nb is itself a semiring with negation given by (−)(m, n) = (n, m), which we call Z; the positive part is N × {0}. The difference from the construction of Z from N, is that here we distinguish (m, n) from (m + k, n + k).

Ab also possesses a negation map given by the “switch”

(−)(a0, a1) = (a1, a0).

The quasi-zeros have the form (a, a). Ab also possesses a negation map given by the “switch”

(−)(a0, a1) = (a1, a0).

The quasi-zeros have the form (a, a). In particular, Nb is itself a semiring with negation given by (−)(m, n) = (n, m), which we call Z; the positive part is N × {0}. The difference from the construction of Z from N, is that here we distinguish (m, n) from (m + k, n + k). IMPORTANT: There is a big difference in taking a + b when a = (−)b ∈ T , in which case it is a◦, as opposed to when a 6= (−)b. This is why we exclude quasi-negatives from the criterion for (−)-bipotence.

(−)-Bipotent triples

Another key property: A triple (A, T , (−)) is (−)-bipotent if a + b ∈ {a, b} whenever a, b ∈ T with b 6= (−)a. (−)-Bipotent triples

Another key property: A triple (A, T , (−)) is (−)-bipotent if a + b ∈ {a, b} whenever a, b ∈ T with b 6= (−)a. IMPORTANT: There is a big difference in taking a + b when a = (−)b ∈ T , in which case it is a◦, as opposed to when a 6= (−)b. This is why we exclude quasi-negatives from the criterion for (−)-bipotence. Two important elements: e =1 ◦ =1( −)1, e0 = e +1 . In tropical applications, e0 = e.

Metatangible triples

The following crucial property, weaker than (−)-bipotence, usually is enough to carry out the theory: A metatangible triple is a triple satisfying unique quasi-negatives, in which a + a0 ∈ T for any a 6= (−)a0 in T . Metatangible triples

The following crucial property, weaker than (−)-bipotence, usually is enough to carry out the theory: A metatangible triple is a triple satisfying unique quasi-negatives, in which a + a0 ∈ T for any a 6= (−)a0 in T . Two important elements: e =1 ◦ =1( −)1, e0 = e +1 . In tropical applications, e0 = e. Our structure of choice, a T -system, thus is a quadruple (A, T , (−), ), where (A, T , (−)) is a T -triple and  is a “surpassing relation” satisfying the crucial property that if a + b  0 for a, b ∈ T then b = (−)a. (This is a bit stronger than unique quasi-negatives.) The most common example of  is ◦, defined above by

◦ a ◦ b iff b = a + c for some c

We now go through different systems which arise in tropical mathematics.

Systems

Semifields are too broad to yield the structural results that we would like, which is the reason that “surpassing relations” and “negation maps” have been introduced. The most common example of  is ◦, defined above by

◦ a ◦ b iff b = a + c for some c

We now go through different systems which arise in tropical mathematics.

Systems

Semifields are too broad to yield the structural results that we would like, which is the reason that “surpassing relations” and “negation maps” have been introduced. Our structure of choice, a T -system, thus is a quadruple (A, T , (−), ), where (A, T , (−)) is a T -triple and  is a “surpassing relation” satisfying the crucial property that if a + b  0 for a, b ∈ T then b = (−)a. (This is a bit stronger than unique quasi-negatives.) Systems

Semifields are too broad to yield the structural results that we would like, which is the reason that “surpassing relations” and “negation maps” have been introduced. Our structure of choice, a T -system, thus is a quadruple (A, T , (−), ), where (A, T , (−)) is a T -triple and  is a “surpassing relation” satisfying the crucial property that if a + b  0 for a, b ∈ T then b = (−)a. (This is a bit stronger than unique quasi-negatives.) The most common example of  is ◦, defined above by

◦ a ◦ b iff b = a + c for some c

We now go through different systems which arise in tropical mathematics. The classical system

First, in classical algebra, with A = T ∪ {0}, the quasi-negative is the usual negative, which is unique of second kind, and A◦ = {0}. Here a ◦ b iff b = a +0= a, so we have the metatangible T -system (A, T , −, =). Here a◦ = aν, e0 = e =1+1 .

The standard supertropical system

The negation map is merely the identity map 1A. The standard supertropical system is (A, T , 1A, ◦), which is of first kind. This is our main model for the tropical theory, and appears in many guises, to be discussed. The standard supertropical system

The negation map is merely the identity map 1A. The standard supertropical system is (A, T , 1A, ◦), which is of first kind. This is our main model for the tropical theory, and appears in many guises, to be discussed. Here a◦ = aν, e0 = e =1+1 . Note that Ab = {(c, c): c ∈ A}.

The symmetrized system is defined as (Ab, Tb, (−), ◦), and is also of tropical type. 0 0 0 0 Here for b = (b0, b1) and b = (b0, b1) we have b ◦ b iff there is 0 0 c ∈ A for which b0 = b1 + c and b0 = b1 + c. In this way we have an important functor from T -modules A (spanned by T ) to symmetrized systems. The symmetrized system is defined as (Ab, Tb, (−), ◦), and is also of tropical type. 0 0 0 0 Here for b = (b0, b1) and b = (b0, b1) we have b ◦ b iff there is 0 0 c ∈ A for which b0 = b1 + c and b0 = b1 + c. In this way we have an important functor from T -modules A (spanned by T ) to symmetrized systems. Note that Ab = {(c, c): c ∈ A}. We always think of
in terms of addition. Note that repeated addition in the hyper-semigroup need not be defined until one passes to P(T ), which makes it difficult to check basic universal identities such as associativity.

Hyper-semigroups

P(T ) denotes the power set, the set of subsets of T . A hyper-semigroup is (T ,
, 0), where I
is a commutative binary operation T × T → P(T ), which also is associative in the sense that if we define

a
S = ∪s∈S a
s,

then (a1
a2)
a3 = a1
(a2
a3) for all ai in T . I 0 is the neutral element.‘ Hyper-semigroups

P(T ) denotes the power set, the set of subsets of T . A hyper-semigroup is (T ,
, 0), where I
is a commutative binary operation T × T → P(T ), which also is associative in the sense that if we define

a
S = ∪s∈S a
s,

then (a1
a2)
a3 = a1
(a2
a3) for all ai in T . I 0 is the neutral element.‘ We always think of
in terms of addition. Note that repeated addition in the hyper-semigroup need not be defined until one passes to P(T ), which makes it difficult to check basic universal identities such as associativity. A hypergroup is a hyper-semigroup (T ,
, 0) for which every element a has a unique hypernegative. Hypernegation induces a negation map on P(T ), via (−)S = {−s : s ∈ S}.

A hypergroup (T ,
, ·, 1) is a hyperfield if (T\ 0, ·, 1) is a group, for which multiplication is distributive over
in T . A good example is to take a multiplicative subgroup G of a field F and let H = (F , ·)/G, with the induced multiplication, and hyperaddition

a
b = {c ∈ F : cG = aG + bG}.

Hyperfields have ties to several subjects including matroid theory.

Hypergroups

A hypernegative of an element a in a hyper-semigroup (T ,
, 0) is an element −a for which0 ∈ a
(−a). Hypernegation induces a negation map on P(T ), via (−)S = {−s : s ∈ S}.

A hypergroup (T ,
, ·, 1) is a hyperfield if (T\ 0, ·, 1) is a group, for which multiplication is distributive over
in T . A good example is to take a multiplicative subgroup G of a field F and let H = (F , ·)/G, with the induced multiplication, and hyperaddition

a
b = {c ∈ F : cG = aG + bG}.

Hyperfields have ties to several subjects including matroid theory.

Hypergroups

A hypernegative of an element a in a hyper-semigroup (T ,
, 0) is an element −a for which0 ∈ a
(−a). A hypergroup is a hyper-semigroup (T ,
, 0) for which every element a has a unique hypernegative. A hypergroup (T ,
, ·, 1) is a hyperfield if (T\ 0, ·, 1) is a group, for which multiplication is distributive over
in T . A good example is to take a multiplicative subgroup G of a field F and let H = (F , ·)/G, with the induced multiplication, and hyperaddition

a
b = {c ∈ F : cG = aG + bG}.

Hyperfields have ties to several subjects including matroid theory.

Hypergroups

A hypernegative of an element a in a hyper-semigroup (T ,
, 0) is an element −a for which0 ∈ a
(−a). A hypergroup is a hyper-semigroup (T ,
, 0) for which every element a has a unique hypernegative. Hypernegation induces a negation map on P(T ), via (−)S = {−s : s ∈ S}. A good example is to take a multiplicative subgroup G of a field F and let H = (F , ·)/G, with the induced multiplication, and hyperaddition

a
b = {c ∈ F : cG = aG + bG}.

Hyperfields have ties to several subjects including matroid theory.

Hypergroups

A hypernegative of an element a in a hyper-semigroup (T ,
, 0) is an element −a for which0 ∈ a
(−a). A hypergroup is a hyper-semigroup (T ,
, 0) for which every element a has a unique hypernegative. Hypernegation induces a negation map on P(T ), via (−)S = {−s : s ∈ S}.

A hypergroup (T ,
, ·, 1) is a hyperfield if (T\ 0, ·, 1) is a group, for which multiplication is distributive over
in T . Hyperfields have ties to several subjects including matroid theory.

Hypergroups

A hypernegative of an element a in a hyper-semigroup (T ,
, 0) is an element −a for which0 ∈ a
(−a). A hypergroup is a hyper-semigroup (T ,
, 0) for which every element a has a unique hypernegative. Hypernegation induces a negation map on P(T ), via (−)S = {−s : s ∈ S}.

A hypergroup (T ,
, ·, 1) is a hyperfield if (T\ 0, ·, 1) is a group, for which multiplication is distributive over
in T . A good example is to take a multiplicative subgroup G of a field F and let H = (F , ·)/G, with the induced multiplication, and hyperaddition

a
b = {c ∈ F : cG = aG + bG}. Hypergroups

A hypernegative of an element a in a hyper-semigroup (T ,
, 0) is an element −a for which0 ∈ a
(−a). A hypergroup is a hyper-semigroup (T ,
, 0) for which every element a has a unique hypernegative. Hypernegation induces a negation map on P(T ), via (−)S = {−s : s ∈ S}.

A hypergroup (T ,
, ·, 1) is a hyperfield if (T\ 0, ·, 1) is a group, for which multiplication is distributive over
in T . A good example is to take a multiplicative subgroup G of a field F and let H = (F , ·)/G, with the induced multiplication, and hyperaddition

a
b = {c ∈ F : cG = aG + bG}.

Hyperfields have ties to several subjects including matroid theory. Let P(T ) denote the additive T -submonoid of P(T ) spanned by T . The surpassing relation now is set inclusion. (P(T ), T , (−), ⊆) is a system, which we call a hypersystem.

Hypersystems

Hypergroups T can be injected naturally into their power sets P(T ), which have a negation map induced from T . (P(T ), T , (−), ⊆) is a system, which we call a hypersystem.

Hypersystems

Hypergroups T can be injected naturally into their power sets P(T ), which have a negation map induced from T . Let P(T ) denote the additive T -submonoid of P(T ) spanned by T . The surpassing relation now is set inclusion. Hypersystems

Hypergroups T can be injected naturally into their power sets P(T ), which have a negation map induced from T . Let P(T ) denote the additive T -submonoid of P(T ) spanned by T . The surpassing relation now is set inclusion. (P(T ), T , (−), ⊆) is a system, which we call a hypersystem. The tropical hypersystem

J Define R∞ = R ∪ {−∞} and define the product a b := a + b and ( max(a, b) if a 6= b, a
b = {c : c ≤ a} if a = b. Thus 0 is the multiplicative identity, −∞ is the additive identity, and we have a hyperfield easily seen to be isomorphic (as semirings) to the standard supertropical semifield, where we identify {c : c ≤ a} with a◦. (In other words, a◦ spells out all possibilities for a + a.) The Krasner (or characteristic 1) hypersystem

Let K = {0; 1} with the usual operations of Boolean algebra, except that now 1
1 = {0; 1}. Again, this generates a sub-semiring of P(K) having three elements, and is just the supertropical algebra system of the trivial monoid T = {1}, where we identify {0; 1} with 1ν. The four elements {{0}, {−1}, {1}, S} constitute the sub-semiring Se of P(S), and comprises a metatangible system isomorphic to the symmetrization of {0, 1}, with {1} 7→ (1, 0), {−1} 7→ (0, 1) and S 7→ (1, 1). The system of signs also does not require the use of hyperfields for its presentation: Write ∞ as an additively absorbing element (x + ∞ = ∞) and put A = {{0}, {−1}, {1}, ∞} with T = {±1}. Addition is idempotent, and 1 + (−)1 = ∞.

The system of signs

Let S := {0, 1, −1} with the usual multiplication, and with hyperaddition defined by 1
1 = {1}, −1
−1 = {−1}, x
0 = 0
x = {x}, and 1
−1 = −1
1 = {0, 1, −1} = S. The system of signs also does not require the use of hyperfields for its presentation: Write ∞ as an additively absorbing element (x + ∞ = ∞) and put A = {{0}, {−1}, {1}, ∞} with T = {±1}. Addition is idempotent, and 1 + (−)1 = ∞.

The system of signs

Let S := {0, 1, −1} with the usual multiplication, and with hyperaddition defined by 1
1 = {1}, −1
−1 = {−1}, x
0 = 0
x = {x}, and 1
−1 = −1
1 = {0, 1, −1} = S. The four elements {{0}, {−1}, {1}, S} constitute the sub-semiring Se of P(S), and comprises a metatangible system isomorphic to the symmetrization of {0, 1}, with {1} 7→ (1, 0), {−1} 7→ (0, 1) and S 7→ (1, 1). The system of signs

Let S := {0, 1, −1} with the usual multiplication, and with hyperaddition defined by 1
1 = {1}, −1
−1 = {−1}, x
0 = 0
x = {x}, and 1
−1 = −1
1 = {0, 1, −1} = S. The four elements {{0}, {−1}, {1}, S} constitute the sub-semiring Se of P(S), and comprises a metatangible system isomorphic to the symmetrization of {0, 1}, with {1} 7→ (1, 0), {−1} 7→ (0, 1) and S 7→ (1, 1). The system of signs also does not require the use of hyperfields for its presentation: Write ∞ as an additively absorbing element (x + ∞ = ∞) and put A = {{0}, {−1}, {1}, ∞} with T = {±1}. Addition is idempotent, and 1 + (−)1 = ∞. Another hypersystem which is not metatangible comes from Viro’s + “triangle” hyperfield, built on R , where a
b = {c : There is a Euclidean triangle of lengths a, b, c.} This is associative since ((a
b)
c) and a
(b
c) are both {d : There is a quadrilateral of lengths a, b, c, d}, but is not (doubly) distributive. The negation map is the identity.

The phase hypersystem and triangle hypersystem

One takes T to be the unit circle, together with its center0. Define a − a = {a, −a, 0}, but the sum of two distinct non-antipodal points is the shortest closed arc joining them. Thus (1 + i) + (1 − i) is the higher closed arc from −i to i. Negation is defined naturally. This system clearly is not metatangible, but has unique negation (the antipode). This is associative since ((a
b)
c) and a
(b
c) are both {d : There is a quadrilateral of lengths a, b, c, d}, but is not (doubly) distributive. The negation map is the identity.

The phase hypersystem and triangle hypersystem

One takes T to be the unit circle, together with its center0. Define a − a = {a, −a, 0}, but the sum of two distinct non-antipodal points is the shortest closed arc joining them. Thus (1 + i) + (1 − i) is the higher closed arc from −i to i. Negation is defined naturally. This system clearly is not metatangible, but has unique negation (the antipode). Another hypersystem which is not metatangible comes from Viro’s + “triangle” hyperfield, built on R , where a
b = {c : There is a Euclidean triangle of lengths a, b, c.} The phase hypersystem and triangle hypersystem

One takes T to be the unit circle, together with its center0. Define a − a = {a, −a, 0}, but the sum of two distinct non-antipodal points is the shortest closed arc joining them. Thus (1 + i) + (1 − i) is the higher closed arc from −i to i. Negation is defined naturally. This system clearly is not metatangible, but has unique negation (the antipode). Another hypersystem which is not metatangible comes from Viro’s + “triangle” hyperfield, built on R , where a
b = {c : There is a Euclidean triangle of lengths a, b, c.} This is associative since ((a
b)
c) and a
(b
c) are both {d : There is a quadrilateral of lengths a, b, c, d}, but is not (doubly) distributive. The negation map is the identity. When (A, T , (−), ) is a system, (Fun(S, A)fin, Fun(S, T ), (−), ) is a system, where Fun(S, A)fin, is the semiring of finite sums from Fun(S, T ). (The polynomial system) As a special case, given a set Λ = {λi : i ∈ I } of commuting indeterminates, there is a system (A[Λ], TA[Λ], (−), ), where TA[Λ] is the set of monomials with coefficients in T .

Function systems

Suppose A has a negation map (−) and surpassing relation . Fun(S, A) has (−) and  defined elementwise, i.e., ((−)f )(s) := (−)(f (s)), and f  g iff f (s)  g(s) for all s ∈ S. (The polynomial system) As a special case, given a set Λ = {λi : i ∈ I } of commuting indeterminates, there is a system (A[Λ], TA[Λ], (−), ), where TA[Λ] is the set of monomials with coefficients in T .

Function systems

Suppose A has a negation map (−) and surpassing relation . Fun(S, A) has (−) and  defined elementwise, i.e., ((−)f )(s) := (−)(f (s)), and f  g iff f (s)  g(s) for all s ∈ S.

When (A, T , (−), ) is a system, (Fun(S, A)fin, Fun(S, T ), (−), ) is a system, where Fun(S, A)fin, is the semiring of finite sums from Fun(S, T ). Function systems

Suppose A has a negation map (−) and surpassing relation . Fun(S, A) has (−) and  defined elementwise, i.e., ((−)f )(s) := (−)(f (s)), and f  g iff f (s)  g(s) for all s ∈ S.

When (A, T , (−), ) is a system, (Fun(S, A)fin, Fun(S, T ), (−), ) is a system, where Fun(S, A)fin, is the semiring of finite sums from Fun(S, T ). (The polynomial system) As a special case, given a set Λ = {λi : i ∈ I } of commuting indeterminates, there is a system (A[Λ], TA[Λ], (−), ), where TA[Λ] is the set of monomials with coefficients in T . Theorems obtained in this setting include a supertropical version of the Nullstellensatz and B´ezout’s Theorem

Roots of functions

Definition: A systemic root of a function f ∈ (Fun(S, A), TFun(S,A), (−), ) is an element s ∈ S for which f (s)  0. Roots of functions

Definition: A systemic root of a function f ∈ (Fun(S, A), TFun(S,A), (−), ) is an element s ∈ S for which f (s)  0.

Theorems obtained in this setting include a supertropical version of the Nullstellensatz and B´ezout’s Theorem In contrast to the classical situation, matrices over systems cannot be factored into diagonal and elementary matrices. The obstruction, introduced by Niv, is studied in joint work with Niv and Sergeev.

Matrix systems

One of the most important applications of systems is in matrices and linear algebra. Define the matrix system

n (Mn(A), ∪i,j=1TMn(A \{(0)}, (−), ),

where the TMn(A are the generalized permutation matrices over T . (−) and  are defined according to entries. Matrix systems

One of the most important applications of systems is in matrices and linear algebra. Define the matrix system

n (Mn(A), ∪i,j=1TMn(A \{(0)}, (−), ),

where the TMn(A are the generalized permutation matrices over T . (−) and  are defined according to entries. In contrast to the classical situation, matrices over systems cannot be factored into diagonal and elementary matrices. The obstruction, introduced by Niv, is studied in joint work with Niv and Sergeev. The (−)-determinant

We write (−)k for (−) applied k times. The (−)-determinant |A| of an n × n matrix A = (ai,j ) defined over an arbitrary (multiplicatively) commutative semiring triple A is

X sgn π (−) aπ, (3)

π∈Sn Qn where aπ := i=1 ai,π(i). The strong transfer principle

Zeilberger showed that a number of determinantal identities admit bijective proofs, and thus have semiring analogs, as noted by Reutenauer and Straubing. Determinantal identities generally have versions which are valid in semirings with a negation map, and at times the result can be strengthened by a counting argument. Furthermore the proofs often pair off mutually negated terms, which thus are in A◦ in the system. This led Gaubert and Guterman to formulate their strong transfer principle, that semiring expressions f = g holding identically on Mn(Z) also hold on Mn(N), and thus on Mn(A) for any semiring A. But passing to the symmetrization, we conclude that f (−)g  0 holds identically on Mn(A). Here are some applications: 0 Write ai,j for the (−)-determinant of the j, i minor of a matrix A. 0 The (−)-adjoint matrix adj(A) is (ai,j ). Laplace’s well-known identity holds over any commutative semiring triple, for any given i: Pn i+j 0 |A| = j=1(−) ai,j ai,j .

Theorem: |A||B|  |AB| for any matrices AB. Theorem: |A||B|  |AB| for any matrices AB.

0 Write ai,j for the (−)-determinant of the j, i minor of a matrix A. 0 The (−)-adjoint matrix adj(A) is (ai,j ). Laplace’s well-known identity holds over any commutative semiring triple, for any given i: Pn i+j 0 |A| = j=1(−) ai,j ai,j . The systemic Cayley-Hamilton Theorem: pA(A)  (0). Theorem: Every -eigenvalue of A is a root of its -characteristic polynomial. The proofs are seen either by the transfer principle, or by pairing off superfluous terms in graphs. This is the method for proving many results about -eigenvalues.

(preceq)-Eigenvalues and the systemic Cayley-Hamilton Theorem

A -eigenvector of a matrix A with -eigenvalue α is a vector v such that Av  αv. The -characteristic polynomial pA of A is |λI − A|, where I is the identity matrix. Theorem: Every -eigenvalue of A is a root of its -characteristic polynomial. The proofs are seen either by the transfer principle, or by pairing off superfluous terms in graphs. This is the method for proving many results about -eigenvalues.

(preceq)-Eigenvalues and the systemic Cayley-Hamilton Theorem

A -eigenvector of a matrix A with -eigenvalue α is a vector v such that Av  αv. The -characteristic polynomial pA of A is |λI − A|, where I is the identity matrix.

The systemic Cayley-Hamilton Theorem: pA(A)  (0). The proofs are seen either by the transfer principle, or by pairing off superfluous terms in graphs. This is the method for proving many results about -eigenvalues.

(preceq)-Eigenvalues and the systemic Cayley-Hamilton Theorem

A -eigenvector of a matrix A with -eigenvalue α is a vector v such that Av  αv. The -characteristic polynomial pA of A is |λI − A|, where I is the identity matrix.

The systemic Cayley-Hamilton Theorem: pA(A)  (0). Theorem: Every -eigenvalue of A is a root of its -characteristic polynomial. (preceq)-Eigenvalues and the systemic Cayley-Hamilton Theorem

A -eigenvector of a matrix A with -eigenvalue α is a vector v such that Av  αv. The -characteristic polynomial pA of A is |λI − A|, where I is the identity matrix.

The systemic Cayley-Hamilton Theorem: pA(A)  (0). Theorem: Every -eigenvalue of A is a root of its -characteristic polynomial. The proofs are seen either by the transfer principle, or by pairing off superfluous terms in graphs. This is the method for proving many results about -eigenvalues. In particular, if A is nonsingular and T is a group, then |A| is invertible in T . The submatrix rank of a nonsingular matrix A is the largest size of a nonsingular square submatrix of A. Surprise: This need not be the row rank, as evidenced by the matrix + + − + + − + + (4) − + + + over the system of signs, writing + for +1 and − for −1. It has row rank 3, but every square submatrix is singular.

Definition: A matrix A is nonsingular if |A| ∈ T . The submatrix rank of a nonsingular matrix A is the largest size of a nonsingular square submatrix of A. Surprise: This need not be the row rank, as evidenced by the matrix + + − + + − + + (4) − + + + over the system of signs, writing + for +1 and − for −1. It has row rank 3, but every square submatrix is singular.

Definition: A matrix A is nonsingular if |A| ∈ T .

In particular, if A is nonsingular and T is a group, then |A| is invertible in T . Surprise: This need not be the row rank, as evidenced by the matrix + + − + + − + + (4) − + + + over the system of signs, writing + for +1 and − for −1. It has row rank 3, but every square submatrix is singular.

Definition: A matrix A is nonsingular if |A| ∈ T .

In particular, if A is nonsingular and T is a group, then |A| is invertible in T . The submatrix rank of a nonsingular matrix A is the largest size of a nonsingular square submatrix of A. Definition: A matrix A is nonsingular if |A| ∈ T .

In particular, if A is nonsingular and T is a group, then |A| is invertible in T . The submatrix rank of a nonsingular matrix A is the largest size of a nonsingular square submatrix of A. Surprise: This need not be the row rank, as evidenced by the matrix + + − + + − + + (4) − + + + over the system of signs, writing + for +1 and − for −1. It has row rank 3, but every square submatrix is singular. Differences between supertropical and symmetrized systems

The reason is that the previous example fails is that the sum of the rows has entries1 − 1+1 , which is surpasses0 in the supertropical case, but is nonzero. Likewise, the supertropical system satisfies the Frobenius surpassing identity (a + b)n  an + bn, whereas the symmetrized system does not. The tensor product fits in with several well-known constructions, including polynomial systems and matrix systems. ⊗i One can construct the tensor semialgebra ⊕i≥1M , which, as in classical theory, yields a host of related structures.

Tensor products

The tensor product is a very well studied construction in category theory, and in particular for systemic modules over semirings. We get a system by defining T (M ⊗ N ) to be the simple tensors a ⊗ a0 where a ∈ T (M) and a0 ∈ T (N ). ⊗i One can construct the tensor semialgebra ⊕i≥1M , which, as in classical theory, yields a host of related structures.

Tensor products

The tensor product is a very well studied construction in category theory, and in particular for systemic modules over semirings. We get a system by defining T (M ⊗ N ) to be the simple tensors a ⊗ a0 where a ∈ T (M) and a0 ∈ T (N ). The tensor product fits in with several well-known constructions, including polynomial systems and matrix systems. Tensor products

The tensor product is a very well studied construction in category theory, and in particular for systemic modules over semirings. We get a system by defining T (M ⊗ N ) to be the simple tensors a ⊗ a0 where a ∈ T (M) and a0 ∈ T (N ). The tensor product fits in with several well-known constructions, including polynomial systems and matrix systems. ⊗i One can construct the tensor semialgebra ⊕i≥1M , which, as in classical theory, yields a host of related structures. We can work directly with T (V ), defining TT (V ) to be the simple tensors in the tangible vectors of V The systemic approach can be applied to the set G≥2 of tensors of length ≥ 2, by imposing the negation map vw = (−)wv for i > j. This can be accomplished for the free module V by setting ej ei = (−)ei ej for all j > i and extended via distributivity. Then we have (−)(v1 ··· vt ) = ((−)v1)v2 ··· vt . (5)

Gatto observed that in fact T (V ) can be embedded into EndG≥2, and then we have a negation map given by (−)v : w 7→ wv.

Exterior (exterior) semialgebra systems

We start with the tensor semialgebra G := T (V ) over a module V . Gatto observed that in fact T (V ) can be embedded into EndG≥2, and then we have a negation map given by (−)v : w 7→ wv.

Exterior (exterior) semialgebra systems

We start with the tensor semialgebra G := T (V ) over a module V .

We can work directly with T (V ), defining TT (V ) to be the simple tensors in the tangible vectors of V The systemic approach can be applied to the set G≥2 of tensors of length ≥ 2, by imposing the negation map vw = (−)wv for i > j. This can be accomplished for the free module V by setting ej ei = (−)ei ej for all j > i and extended via distributivity. Then we have (−)(v1 ··· vt ) = ((−)v1)v2 ··· vt . (5) Exterior (exterior) semialgebra systems

We start with the tensor semialgebra G := T (V ) over a module V .

We can work directly with T (V ), defining TT (V ) to be the simple tensors in the tangible vectors of V The systemic approach can be applied to the set G≥2 of tensors of length ≥ 2, by imposing the negation map vw = (−)wv for i > j. This can be accomplished for the free module V by setting ej ei = (−)ei ej for all j > i and extended via distributivity. Then we have (−)(v1 ··· vt ) = ((−)v1)v2 ··· vt . (5)

Gatto observed that in fact T (V ) can be embedded into EndG≥2, and then we have a negation map given by (−)v : w 7→ wv. G≥2 has the natural negation map. The appropriate exterior T -triple is (G≥2, T , (−)), where T = {v1 ··· vt : vi ∈ T , t ∈ N}. Note that we have a negation map on G≥2 even without a negation map on V . Furthermore, we did not need to take homomorphic ◦ images. The quasi-zeros G≥2 are spanned by the v ⊗ v : v ∈ V .

To obtain exterior semialgebras, we follow the familiar construction of the exterior algebra over a module V , as the tensor algebra of V modulo the exterior relation. If V is spanned by {vi : i ∈ I }, then to verify the exterior relation it is enough to check that

vi vj = (−)vj vi , ∀i, j ∈ I .

π π Thus vπ(1) ··· vπ(t) = (−) v1 ··· vt , where (−) denotes the sign of the permutation. To obtain exterior semialgebras, we follow the familiar construction of the exterior algebra over a module V , as the tensor algebra of V modulo the exterior relation. If V is spanned by {vi : i ∈ I }, then to verify the exterior relation it is enough to check that

vi vj = (−)vj vi , ∀i, j ∈ I .

π π Thus vπ(1) ··· vπ(t) = (−) v1 ··· vt , where (−) denotes the sign of the permutation. G≥2 has the natural negation map. The appropriate exterior T -triple is (G≥2, T , (−)), where T = {v1 ··· vt : vi ∈ T , t ∈ N}. Note that we have a negation map on G≥2 even without a negation map on V . Furthermore, we did not need to take homomorphic ◦ images. The quasi-zeros G≥2 are spanned by the v ⊗ v : v ∈ V . The system (on G≥2) is rounded out with the following surpassing relation. Definition: The exterior surpassing relation  on G≥2 is given by: 2 I ei  0. ◦ I a  b if b = a + d for some d ∈ G≥2 . The exterior surpassing relation restricts to equality on T . This has been used with Gatto to derive semialgebra versions of standard results from the classical theory, which generalize the Cayley-Hamilton theorem. In a system define the elements 1 =1 , and inductively n + 1 = n +1. Any metatangible system (A, T , (−), ) over a group T must satisfy one of the following: 0 I (−) is of the first kind. A = ∪m∈N mT , and e =3 . I 3=2 , the supertropical system. I 3 6=1 . Then T is (−)-bipotent, and (A, T , (−), ) is isomorphic to a “layered system” (layered by N in characteristic 0, and Z/k in characteristic k > 0). I 3=1 . Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but there are other examples.

Characterizing metatangible systems

One can describe pretty well the metatangible systems, even though we lack a full characterization. Any metatangible system (A, T , (−), ) over a group T must satisfy one of the following: 0 I (−) is of the first kind. A = ∪m∈N mT , and e =3 . I 3=2 , the supertropical system. I 3 6=1 . Then T is (−)-bipotent, and (A, T , (−), ) is isomorphic to a “layered system” (layered by N in characteristic 0, and Z/k in characteristic k > 0). I 3=1 . Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but there are other examples.

Characterizing metatangible systems

One can describe pretty well the metatangible systems, even though we lack a full characterization. In a system define the elements 1 =1 , and inductively n + 1 = n +1. 0 I (−) is of the first kind. A = ∪m∈N mT , and e =3 . I 3=2 , the supertropical system. I 3 6=1 . Then T is (−)-bipotent, and (A, T , (−), ) is isomorphic to a “layered system” (layered by N in characteristic 0, and Z/k in characteristic k > 0). I 3=1 . Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but there are other examples.

Characterizing metatangible systems

One can describe pretty well the metatangible systems, even though we lack a full characterization. In a system define the elements 1 =1 , and inductively n + 1 = n +1. Any metatangible system (A, T , (−), ) over a group T must satisfy one of the following: I 3=2 , the supertropical system. I 3 6=1 . Then T is (−)-bipotent, and (A, T , (−), ) is isomorphic to a “layered system” (layered by N in characteristic 0, and Z/k in characteristic k > 0). I 3=1 . Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but there are other examples.

Characterizing metatangible systems

One can describe pretty well the metatangible systems, even though we lack a full characterization. In a system define the elements 1 =1 , and inductively n + 1 = n +1. Any metatangible system (A, T , (−), ) over a group T must satisfy one of the following: 0 I (−) is of the first kind. A = ∪m∈N mT , and e =3 . I 3=1 . Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but there are other examples.

Characterizing metatangible systems

One can describe pretty well the metatangible systems, even though we lack a full characterization. In a system define the elements 1 =1 , and inductively n + 1 = n +1. Any metatangible system (A, T , (−), ) over a group T must satisfy one of the following: 0 I (−) is of the first kind. A = ∪m∈N mT , and e =3 . I 3=2 , the supertropical system. I 3 6=1 . Then T is (−)-bipotent, and (A, T , (−), ) is isomorphic to a “layered system” (layered by N in characteristic 0, and Z/k in characteristic k > 0). Characterizing metatangible systems

One can describe pretty well the metatangible systems, even though we lack a full characterization. In a system define the elements 1 =1 , and inductively n + 1 = n +1. Any metatangible system (A, T , (−), ) over a group T must satisfy one of the following: 0 I (−) is of the first kind. A = ∪m∈N mT , and e =3 . I 3=2 , the supertropical system. I 3 6=1 . Then T is (−)-bipotent, and (A, T , (−), ) is isomorphic to a “layered system” (layered by N in characteristic 0, and Z/k in characteristic k > 0). I 3=1 . Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but there are other examples. I (−) is of the second kind, with two main possibilities:

I T is (−)-bipotent, and T (and thus A) is idempotent. Taking the congruence identifying a with (−), A/ ≡ is a (−)-bipotent system of the first kind, under the induced addition and multiplication. ◦ I T is not (−)-bipotent. Then3= 1. Hence A = T ∩ T . Either N ⊆ T , or (A, T , −, ) has characteristic k for some k ≥ 1. In the latter case, (A, T , −, ) is layered by Z/k. These include “layered semirings,” of the form A = L × G, where L is the “layering semiring,” which has its own negation map that we designate as −, and (G, ·) is an ordered monoid. Another example is the truncated algebra at n, where addition and multiplication are given by identifying every number greater than n with n. In other words,

k1 + k2 = n in L if k1 + k2 ≥ n in N;

k1k2 = n in L if k1k2 ≥ n in N.

In order to have a chance to classify (−)-bipotent (or more generally metatangible) systems, we need a full list of examples. Another example is the truncated algebra at n, where addition and multiplication are given by identifying every number greater than n with n. In other words,

k1 + k2 = n in L if k1 + k2 ≥ n in N;

k1k2 = n in L if k1k2 ≥ n in N.

In order to have a chance to classify (−)-bipotent (or more generally metatangible) systems, we need a full list of examples. These include “layered semirings,” of the form A = L × G, where L is the “layering semiring,” which has its own negation map that we designate as −, and (G, ·) is an ordered monoid. In order to have a chance to classify (−)-bipotent (or more generally metatangible) systems, we need a full list of examples. These include “layered semirings,” of the form A = L × G, where L is the “layering semiring,” which has its own negation map that we designate as −, and (G, ·) is an ordered monoid. Another example is the truncated algebra at n, where addition and multiplication are given by identifying every number greater than n with n. In other words,

k1 + k2 = n in L if k1 + k2 ≥ n in N;

k1k2 = n in L if k1k2 ≥ n in N. A -morphism

0 0 0 0 ϕ :(M, TM, (−), ) → (M , TM0 , (−) ,  ) is a map ϕ : M → M0 satisfying the following properties for 0 ai ∈ T and b  b in M, bi in M: I ϕ(0) =0 .

I ϕ((−)b1) = (−)ϕ(b1); 0 I ϕ(b1 + b2)  ϕ(b1) + ϕ(b2); 0 I ϕ(a1b)  a1ϕ(b). 0 0 I ϕ(b)  ϕ(b ). I An interesting example comes from valuation theory. Valuations can be displayed as -morphisms of semirings, writing the (ordered semigroup) target of the valuation as a semiring (using multiplicative notation instead of additive notation) as we did with the max-plus semiring.

I In supertropical mathematics, f satisfies

◦ f (b1 + b2) + b = f (b1) + f (b2); (6)

thus either f (b1 + b2) = f (b1) + f (b2), or f (b1) = f (b2). In particular, as a special case of (1), tropicalization of Puiseux series via the Puiseux valuation is a -morphism, and provides a major motivation in introducing -morphisms.

Examples of -morphisms

Let us describe -morphisms for our examples, to see why we want to consider -morphisms rather than merely homomorphisms. I In supertropical mathematics, f satisfies

◦ f (b1 + b2) + b = f (b1) + f (b2); (6)

thus either f (b1 + b2) = f (b1) + f (b2), or f (b1) = f (b2). In particular, as a special case of (1), tropicalization of Puiseux series via the Puiseux valuation is a -morphism, and provides a major motivation in introducing -morphisms.

Examples of -morphisms

Let us describe -morphisms for our examples, to see why we want to consider -morphisms rather than merely homomorphisms.

I An interesting example comes from valuation theory. Valuations can be displayed as -morphisms of semirings, writing the (ordered semigroup) target of the valuation as a semiring (using multiplicative notation instead of additive notation) as we did with the max-plus semiring. Examples of -morphisms

Let us describe -morphisms for our examples, to see why we want to consider -morphisms rather than merely homomorphisms.

I An interesting example comes from valuation theory. Valuations can be displayed as -morphisms of semirings, writing the (ordered semigroup) target of the valuation as a semiring (using multiplicative notation instead of additive notation) as we did with the max-plus semiring.

I In supertropical mathematics, f satisfies

◦ f (b1 + b2) + b = f (b1) + f (b2); (6)

thus either f (b1 + b2) = f (b1) + f (b2), or f (b1) = f (b2). In particular, as a special case of (1), tropicalization of Puiseux series via the Puiseux valuation is a -morphism, and provides a major motivation in introducing -morphisms. I For fuzzy rings, a fuzzy homomorphism

f :(K; +; ×, εK , K0) → (L; +; ×; εL; L0)

of fuzzy rings is defined as satisfying: For any × Pn Pn {a1,..., an} ∈ K if i=1 ai ∈ K0 then i=1 f (ai ) ∈ L0. Any -morphism in our setting is a fuzzy homomorphism since L0 Pn Pn is an ideal, and thus i=1 f (ai ) ∈ f ( i=1 ai ) + L0 = L0. The other direction might not hold. The same reasoning holds for the tracts of Baker and Bowler.

I For hypersystems, a -morphism f satisfies

f (b1
b2) ⊆ f (b1)
f (b2), (7) the definition used in the hyperfield literature. I For hypersystems, a -morphism f satisfies

f (b1
b2) ⊆ f (b1)
f (b2), (7) the definition used in the hyperfield literature.

I For fuzzy rings, a fuzzy homomorphism

f :(K; +; ×, εK , K0) → (L; +; ×; εL; L0)

of fuzzy rings is defined as satisfying: For any × Pn Pn {a1,..., an} ∈ K if i=1 ai ∈ K0 then i=1 f (ai ) ∈ L0. Any -morphism in our setting is a fuzzy homomorphism since L0 Pn Pn is an ideal, and thus i=1 f (ai ) ∈ f ( i=1 ai ) + L0 = L0. The other direction might not hold. The same reasoning holds for the tracts of Baker and Bowler. One has basic notions such as -idempotent (a2  a), -central (zr(−)rz  0 for all r), -annihilator ( ar  0), which immediately give rise to analogs of ring-theoretic notions, such as a -Peirce decomposition. How far can one push these, starting with Wedderburn’s theorems?

Elementary ()-semiring theory

Here are some promising areas for further research. for further research. Elementary ()-semiring theory

Here are some promising areas for further research. for further research. One has basic notions such as -idempotent (a2  a), -central (zr(−)rz  0 for all r), -annihilator ( ar  0), which immediately give rise to analogs of ring-theoretic notions, such as a -Peirce decomposition. How far can one push these, starting with Wedderburn’s theorems? Definition: A semiring polynomial identity (semiring PI) of a semiring A is a pair (f (x1,..., xm), g(x1,..., xn)) for which (m) f (r1,..., rm) = g(x1,..., xn), ∀r = (r1,..., rm,..., rn) ∈ A .

Polynomial identities of semirings

We assume that every knows the definition of a polynomial identity (PI) of a ring. For semirings in order to bypass negation, one can use the alternative definition from universal algebra: Polynomial identities of semirings

We assume that every knows the definition of a polynomial identity (PI) of a ring. For semirings in order to bypass negation, one can use the alternative definition from universal algebra:

Definition: A semiring polynomial identity (semiring PI) of a semiring A is a pair (f (x1,..., xm), g(x1,..., xn)) for which (m) f (r1,..., rm) = g(x1,..., xn), ∀r = (r1,..., rm,..., rn) ∈ A . It is easy to see that when A happens to be a ring, f (x1,..., xm), g(x1,..., xn) is a semiring PI of A if and only if f − g is a PI of A as a ring. The “transfer principle” shows that if f − g is a multilinear PI of the matrix algebra Mn(Z) then (f , g) is a semiring PI of Mn(Gmax), for any ordered group G. In the other direction, the Frobenius surpassing identity of Zmax is not an identity of Z. Even more extreme, Izhakian and Margolis produced nontrivial semigroup identities of Mn(Zmax), but there do not exist semigroup identities of the ring Mn(Z) for n ≥ 2. This shows that the tropical PI-theory is richer than the ring-theoretic PI-theory. But what about semirings in general? In view of the Izhakian-Margolis result, it seems unlikely, but I do not know of a specific example. But using symmetrization one can recast these questions to systems.

Arguing in this way, one can solve Specht’s problem for semiring PIs for rings, viewed as semirings, that all semiring PI’s of a ring is a consequence of finitely many semiring PI’s. Arguing in this way, one can solve Specht’s problem for semiring PIs for rings, viewed as semirings, that all semiring PI’s of a ring is a consequence of finitely many semiring PI’s. But what about semirings in general? In view of the Izhakian-Margolis result, it seems unlikely, but I do not know of a specific example. But using symmetrization one can recast these questions to systems. Develop the SPI-theory, based on the SPI’s of matrices. Amusing observation: the supertropical Amitsur-Levitzki theorem has an almost trivial graph-theoretical proof, since the relevant cycles repeat. Question: Does Specht’s problem have a positive solution for PI-systems, perhaps along the lines of Kemer’s celebrated solution (Most likely one needs to assume finitely generated)? Question: (The systemic version of Shirshov’s Theorem) Is every PI-system finitely generated and integral over a semiring system, finitely spanned

Definition: A systemic polynomial identity (SPI) of a system (A, T , G, ν) is a (noncommutative) polynomial f (x1,..., xm) ∈ A[x1,... xm] for (m) which f (r1,..., rm)  0, ∀r = (r1,..., rm) ∈ A . We write idsys(A) for the set of SPI’s of A. Clearly, if (f , g) is a semiring PI then f (−)g is an SPI. Amusing observation: the supertropical Amitsur-Levitzki theorem has an almost trivial graph-theoretical proof, since the relevant cycles repeat. Question: Does Specht’s problem have a positive solution for PI-systems, perhaps along the lines of Kemer’s celebrated solution (Most likely one needs to assume finitely generated)? Question: (The systemic version of Shirshov’s Theorem) Is every PI-system finitely generated and integral over a semiring system, finitely spanned

Definition: A systemic polynomial identity (SPI) of a system (A, T , G, ν) is a (noncommutative) polynomial f (x1,..., xm) ∈ A[x1,... xm] for (m) which f (r1,..., rm)  0, ∀r = (r1,..., rm) ∈ A . We write idsys(A) for the set of SPI’s of A. Clearly, if (f , g) is a semiring PI then f (−)g is an SPI. Develop the SPI-theory, based on the SPI’s of matrices. Question: Does Specht’s problem have a positive solution for PI-systems, perhaps along the lines of Kemer’s celebrated solution (Most likely one needs to assume finitely generated)? Question: (The systemic version of Shirshov’s Theorem) Is every PI-system finitely generated and integral over a semiring system, finitely spanned

Definition: A systemic polynomial identity (SPI) of a system (A, T , G, ν) is a (noncommutative) polynomial f (x1,..., xm) ∈ A[x1,... xm] for (m) which f (r1,..., rm)  0, ∀r = (r1,..., rm) ∈ A . We write idsys(A) for the set of SPI’s of A. Clearly, if (f , g) is a semiring PI then f (−)g is an SPI. Develop the SPI-theory, based on the SPI’s of matrices. Amusing observation: the supertropical Amitsur-Levitzki theorem has an almost trivial graph-theoretical proof, since the relevant cycles repeat. Question: (The systemic version of Shirshov’s Theorem) Is every PI-system finitely generated and integral over a semiring system, finitely spanned

Definition: A systemic polynomial identity (SPI) of a system (A, T , G, ν) is a (noncommutative) polynomial f (x1,..., xm) ∈ A[x1,... xm] for (m) which f (r1,..., rm)  0, ∀r = (r1,..., rm) ∈ A . We write idsys(A) for the set of SPI’s of A. Clearly, if (f , g) is a semiring PI then f (−)g is an SPI. Develop the SPI-theory, based on the SPI’s of matrices. Amusing observation: the supertropical Amitsur-Levitzki theorem has an almost trivial graph-theoretical proof, since the relevant cycles repeat. Question: Does Specht’s problem have a positive solution for PI-systems, perhaps along the lines of Kemer’s celebrated solution (Most likely one needs to assume finitely generated)? Definition: A systemic polynomial identity (SPI) of a system (A, T , G, ν) is a (noncommutative) polynomial f (x1,..., xm) ∈ A[x1,... xm] for (m) which f (r1,..., rm)  0, ∀r = (r1,..., rm) ∈ A . We write idsys(A) for the set of SPI’s of A. Clearly, if (f , g) is a semiring PI then f (−)g is an SPI. Develop the SPI-theory, based on the SPI’s of matrices. Amusing observation: the supertropical Amitsur-Levitzki theorem has an almost trivial graph-theoretical proof, since the relevant cycles repeat. Question: Does Specht’s problem have a positive solution for PI-systems, perhaps along the lines of Kemer’s celebrated solution (Most likely one needs to assume finitely generated)? Question: (The systemic version of Shirshov’s Theorem) Is every PI-system finitely generated and integral over a semiring system, finitely spanned Develop a theory of “classical -semigroups.”

The classical systemic groups

Itzhakian, Niv, and R defined the systemic version of GL(n, A) as ◦ {A ∈ Mn(A): |A| ∈/ A }, and SL as {A ∈ Mn(A): |A|  1}, and proved various properties. One could likewise define the orthogonal groups via AAt  I , etc. The classical systemic groups

Itzhakian, Niv, and R defined the systemic version of GL(n, A) as ◦ {A ∈ Mn(A): |A| ∈/ A }, and SL as {A ∈ Mn(A): |A|  1}, and proved various properties. One could likewise define the orthogonal groups via AAt  I , etc. Develop a theory of “classical -semigroups.” Representation theory

One has the systemic trace. How should representation theory take shape? Presumably it should start with homomorphisms to GL(n, A), and involve module theory. Lie semialgebras

A Lie -semialgebra is an A-module L with a negation map (−), endowed with an A-bilinear -Lie bracket [ ] satisfying, for all x, y ∈ L:

1.[ xy] + [yx]  0L,

2.[ xx]  0L,

3. ad[xy] + ady adx  adx ady , where adx (z) = [xz]. Can one prove Lie’s Theorem and Engel’s Theorem in this context? How do Dynkin diagrams fit in? Grobner-Shirshov Bases

Although it is easy to describe the free associative semialgebra and the free Lie semialgebra, the set of relations of an algebra is a congruence rather than an ideal, so is harder to describe. Furthermore, one wants to take  into account. How would one describe a -Grobner-Shirshov Basis? Develop a systemic version which includes the genus, elliptic curves, and tackle classical theorems such as Riemann-Roch.

Geometry

What is the dimension of a variety? We have algebraic definitions of tropical discriminants and resultants, with their basic properties. Geometry

What is the dimension of a variety? We have algebraic definitions of tropical discriminants and resultants, with their basic properties. Develop a systemic version which includes the genus, elliptic curves, and tackle classical theorems such as Riemann-Roch. Thank you for your attention.