A General Algebraic Structure Theory for Tropical Mathematics Algebra Conference in Spa

A general algebraic structure theory for tropical mathematics Algebra Conference in Spa

Louis Rowen, Bar-Ilan University

Tuesday 20 June, 2017

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 1 / 75 Overview PART I: Overview

Many algebraic theories involve the study of a set T with incomplete structure that can be understood better by embedding T in a larger set A endowed with more structure. We start with the following set-up:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 2 / 75 Overview

A T -module over a set T is an additive monoid (A, +, 0A) together with scalar multiplication T × A → A satisfying distributivity over T in the sense that a(b1 + b2) = ab1 + ab2 for a ∈ T , bi ∈ A, also with the stipulation that a0A = 0A for all a in T .

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 3 / 75 A semiring† is idempotent if a + a = a; it is bipotent if a + b ∈ {a, b}.

We also want to consider modules M over a semiring R with zero 0R , often called semimodules in the literature; one must stipulate that 0R b = 0M , ∀b ∈ M.

Overview Semirings†

Usually A is a semiring. We delete the zero element in the deﬁnition of a † semiring, because it just gets in the way: A semiring (R, +, ·, 1R ) is a set R equipped with binary operations + and · such that: (R, +) is an Abelian semigroup;

(R, · , 1R ) is a monoid with identity element 1R ; Multiplication distributes over addition.

There exist a, b ∈ R such that a + b = 1R . (The last axiom is much weaker than requiring the existence of a zero element.) Thus, we have all the ring axioms except negation. † † A semiﬁeld is a semiring for which (R, · , 1R ) is an Abelian group.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 4 / 75 We also want to consider modules M over a semiring R with zero 0R , often called semimodules in the literature; one must stipulate that 0R b = 0M , ∀b ∈ M.

Overview Semirings†

(R, · , 1R ) is a monoid with identity element 1R ; Multiplication distributes over addition.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 4 / 75 Overview Semirings†

(R, · , 1R ) is a monoid with identity element 1R ; Multiplication distributes over addition.

We also want to consider modules M over a semiring R with zero 0R , often called semimodules in the literature; one must stipulate that 0R b = 0M , ∀b ∈ M.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 4 / 75 Overview Some universal algebra

Universal algebra is a natural general structure theory described in terms of “congruences” which, although more complicated than the usual algebraic structure theory because of the lack of an intrinsic negative, has a wide range of applications. Not every structure involved in tropical mathematics can be put in the framework of signatures in universal algebra. Nevertheless, the language of universal algebra uniﬁes many algebraic theories, including the common tropical theories.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 5 / 75 Overview

Deﬁnition:

A carrier is a collection of sets A1, A2,..., At . A set of operators is a set Ω := ∪m∈NΩ(m) where each Ω(m) := {ωm,j : j ∈ Jm} in turn is a set of formal symbols ωm,j = ωm,j (x1,j ,..., xm,j ) interpreted as a map

ωm,j : Aj1 × · · · × Ajm → Aim,j . Each operator ωm,j , called an (m-ary) operator, has a target index im,j , indicating where the operator takes its values.

We deﬁne a targeted Ω-formula inductively: Each formal letter xu,i is an Ω-formula with target i, and if φ1, . . . , φm are Ω-formulas with respective targets iu,j , 1 ≤ u ≤ m, and if ωm,j (x1,j ,..., xm,j ) ∈ Ω is compatible with φ in the sense that iu,j is the subscript for xu,j for each u, then ωm,j (φ1, . . . , φm) also is an Ω-formula.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 6 / 75 One can formulate T -modules in terms of universal algebra, often with extra structure on T passed on to A. For example T could be a monoid, in which case we also require associativity ((a1a2)b = a1(a2b) for all ai ∈ T and b ∈ A). Or T could have a Lie structure, and A could be a Lie module.

Overview

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 7 / 75 Overview

A universal relation is a pair (φ, ψ) of Ω-formulas (having the same target). It is the way we identify two expressions. (Associativity is a good example of a universal relation.) A signature is a pair (Ω, I), where Ω is a set of operators and I is a set of universal relations. Writing I for the set of universal relations, we also call {A1, A2,..., At } an (Ω; I)-algebra. In our theory, the carrier will include both A and T , and the signature will take their structure into account. One can formulate T -modules in terms of universal algebra, often with extra structure on T passed on to A. For example T could be a monoid, in which case we also require associativity ((a1a2)b = a1(a2b) for all ai ∈ T and b ∈ A). Or T could have a Lie structure, and A could be a Lie module.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 7 / 75 Overview Universal algebra and categories

A signature in universal algebra gives rise to a category C whose objects are the carriers and whose morphisms are functions that preserve the operators, i.e.,

f (ω(a1,..., am)) = ω(f (a1),..., f (am)).

But later we will weaken the deﬁnition of morphism, to make it more appropriate to “systems.”

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 8 / 75 When S is a set, we usually write I instead of S.

The support of f is {s ∈ S : f (s) 6= 0}. In this situation, we can take T to be the morphisms having support of order 1.

Overview Function systems

Given a category C from universal algebra and a small category S, we deﬁne CS to be the category of morphisms from S to C. CS can be seen to have the same signature as C, where operations are deﬁned componentwise, i.e., if fi : S → C, then

ω(f1,..., am)(s) := ω(f1(s),..., fm(s)), ∀s ∈ S.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 9 / 75 The support of f is {s ∈ S : f (s) 6= 0}. In this situation, we can take T to be the morphisms having support of order 1.

Overview Function systems

ω(f1,..., am)(s) := ω(f1(s),..., fm(s)), ∀s ∈ S.

When S is a set, we usually write I instead of S.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 9 / 75 Overview Function systems

ω(f1,..., am)(s) := ω(f1(s),..., fm(s)), ∀s ∈ S.

When S is a set, we usually write I instead of S.

The support of f is {s ∈ S : f (s) 6= 0}. In this situation, we can take T to be the morphisms having support of order 1.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 9 / 75 Overview The convolution product

If the signature has multiplication and (S, +) is a monoid, we often deﬁne instead the convolution product f ∗ g by X (f ∗ g)(s) = f (u)g(v), u+v=s

but this only makes sense when there are only ﬁnitely many u, v with u + v = s. This works for the morphisms of ﬁnite support, which we write as C(S). For example, the polynomials are C(N) with the convolution product.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 10 / 75 Overview Classical examples

In many examples but not all, T is a monoid.

1 A is an integral domain and T = A\{0}; 2 A is a graded algebra, and T is the monoid of homogeneous elements. 3 A is a vector space with base T . 4 More speciﬁcally, A is an algebra with a multiplicative base T . This could be viewed in terms of the previous slide. For example, A could be the group algebra of a group T . 5 A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements).

6 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 table algebras.)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 11 / 75 Overview Two non-classical examples

Our interest however was stimulated by examples outside of classical algebra. Before delving into the theory, we consider two of the main examples, postponing the others until we develop some theory:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 12 / 75 Overview The max-plus algebra

The parent structure in tropical algebra, which also arises in varied contexts in applied mathematics, is the well-known max-plus algebra on an ordered monoid, where multiplication is the old addition, and addition is the maximum. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax. Speciﬁcally, ordered groups, such as (Q, +) or (R, +), are viewed at once as max-plus semiﬁelds†, generalizing to the following elegant observation of Green:

(To emphasize the algebraic structure we still use the usual algebraic notation of · and + throughout.) Any ordered monoid (M, · ) gives rise to a bipotent semiring†, where we deﬁne a + b to be max{a, b}.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 13 / 75 Customarily the target Q of − val has been viewed as the max-plus algebra, 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 v(f + g) = 3 > 2.

Overview Puiseux series and tropicalization

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 14 / 75 For example, if f = 2λ2 + 7λ4 and g = −2λ2 + 5λ3 + 7λ4 then f + g = 5λ3 + 14λ4 and v(f + g) = 3 > 2.

Overview Puiseux series and tropicalization

For a structure A of a given signature in universal algebra, one can deﬁne the set A = A{{t}} of Puiseux series on the variable t, which is the set of P∞ k/N formal series of the form f = k=` ck t where N ∈ N, ` ∈ Z, and ck ∈ S, with the convolution product. Then one has the Puiseux valuation val : A{{t}} \ {0} → Q ⊂ R deﬁned by val(f ) = − min{k/N}, (1) ck 6=0 which we also call tropicalization. Customarily the target Q of − val has been viewed as the max-plus algebra, 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.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 14 / 75 Overview Puiseux series and tropicalization

Thus, valuations behave like the min-plus algebra EXCEPT perhaps when evaluated on elements having the same value. Hence, tropicalization is not functorial! We need a replacement to the max-plus which is almost bipotent, in the sense that a + b = max{a, b} except for a = b.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 15 / 75 (Attention focuses on supertropical semirings†, but the more general deﬁnition of ν-semiring† enables one to work with polynomials and matrices.)

Overview Supertropical semirings† and supertropical domains†

To remedy this, we recall some basics of supertropical algebra. Deﬁnition: A ν-semiring† is a quadruple R := (R, T , G, ν) where R is a semiring†, T is a submonoid, and G ⊂ R is a semiring† ideal, with a multiplicative monoid homomorphism ν : R → G, satisfying ν2 = ν as well as the condition: a + b = ν(a) whenever ν(a) = ν(b). R is called a supertropical semiring† when ν is onto, G is ordered, and

a + b = a whenever ν(a) > ν(b).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 16 / 75 Overview Supertropical semirings† and supertropical domains†

a + b = a whenever ν(a) > ν(b).

(Attention focuses on supertropical semirings†, but the more general deﬁnition of ν-semiring† enables one to work with polynomials and matrices.)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 16 / 75 Overview

The elements of G are called ghost elements and ν : R → G is called the ghost map. T is the monoid of tangible elements, and encapsulates the tropical aspect. A supertropical semiring† R is called a supertropical domain† when the multiplicative monoid (R, · ) is commutative, ν|T is 1:1, and R is cancellative. In this case ν|T : T → G is a monoid isomorphism, and T inherits the order from G.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 17 / 75 Thus, we start with T and pass to the standard supertropical semiﬁeld† A. This is our main model for the tropical theory.

Overview

The standard supertropical semiﬁeld† is A := T ∪ T ν (where customarily T = Qmax or Rmax). Addition is now given by  ν(a) whenever a = b,  a + b = a whenever a > b, b whenever a < b.

The standard supertropical semiﬁeld is the standard supertropical semiﬁeld† with 0 adjoined.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 18 / 75 Overview

The standard supertropical semifield is the standard supertropical semifield† with 0 adjoined. Thus, we start with T and pass to the standard supertropical semifield† A. This is our main model for the tropical theory.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 18 / 75 Overview

Our overall goal in this talk is to provide an algebraic umbrella, especially to tropical mathematics and related areas, in a general framework which includes as much of the classical theory as possible, with the goal of addressing the following basic questions:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 19 / 75 Overview

What is the basic algebraic structure on which to pin our theory? What is a variety in this framework? (We would want an algebraic definition that matches geometric intuition.) Can the definition be made natural, in the sense that it commutes with tropicalization? What is the dimension of a variety? How can we develop linear algebra to obtain analogs of the main theorems of classical matrix theory? How does one algebraically define basic geometric invariants such as resultants, discriminants, genus, etc.? How should representation theory take shape? What are the analogs of the classical groups, exterior algebras, and Lie algebras for example? Is there a version of module theory which handles direct sum decompositions of submodules of free modules, that could support a homological theory?

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 20 / 75 In order to overcome partially the lack of negatives, we introduce a formal negation map a 7→ (−)a which satisﬁes all of the properties of negation except a + ((−)a) = 0. A negation map (−) is an additive homomorphism (−):(A, +) → (A, +) of order ≤ 2, written a 7→ (−)a. When A has multiplication we also require

(−)(a1a2) = ((−)a1)a2 = a1((−)a2).

We view (−) as a unary operator in universal algebra, and require that it preserves the other linear operators.

Overview Negation maps

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 21 / 75 When A has multiplication we also require

(−)(a1a2) = ((−)a1)a2 = a1((−)a2).

We view (−) as a unary operator in universal algebra, and require that it preserves the other linear operators.

Overview Negation maps

There is a basic diﬃculty in developing the structure theory of semirings in place of rings: Cosets need not be disjoint (this fact relying on additive cancelation, which fails in the max-plus algebra, since 1 + 3 = 2 + 3 = 3). In order to overcome partially the lack of negatives, we introduce a formal negation map a 7→ (−)a which satisﬁes all of the properties of negation except a + ((−)a) = 0. A negation map (−) is an additive homomorphism (−):(A, +) → (A, +) of order ≤ 2, written a 7→ (−)a.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 21 / 75 We view (−) as a unary operator in universal algebra, and require that it preserves the other linear operators.

Overview Negation maps

(−)(a1a2) = ((−)a1)a2 = a1((−)a2).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 21 / 75 Overview Negation maps

(−)(a1a2) = ((−)a1)a2 = a1((−)a2).

We view (−) as a unary operator in universal algebra, and require that it preserves the other linear operators.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 21 / 75 Overview

The usual negation in classical algebra is clearly a negation map, but in non-classical situations we lack negatives. In particular, negation is notably absent in the tropical theory, but is circumvented in two main ways: The identity itself is a perfectly valid negation map (since one just erases the minus signs). One can introduce a negation map through the process of “symmetrization,” 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).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 22 / 75 We write a(−)b for a + ((−)b), and a = (±b) when a = b or a = (−)b. Given a ∈ A we deﬁne the quasi-zero a◦ := a(−)a, and

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a. A semigroup (A, +) has characteristic k > 0 if k + 1 = 1 with k ≥ 1 minimal. A has characteristic 0 if A does not have characteristic k for any k ≥ 1. Any idempotent algebra has “characteristic 1,” leading to the notion of “F1 geometry.’

Overview

Define e = 1◦ = 1(−)1, e0 = e + 1. (2) Also we define 1 = 1, and inductively n + 1 = n + 1. The negation map (−) is said to be of the first kind if (−)1 = 1 (and thus (−) is the identity), and of the second kind if (−)a 6= a for all a ∈ T . When we have cancelation, it is enough to check whether or not (−)1 = 1.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 23 / 75 A semigroup (A, +) has characteristic k > 0 if k + 1 = 1 with k ≥ 1 minimal. A has characteristic 0 if A does not have characteristic k for any k ≥ 1. Any idempotent algebra has “characteristic 1,” leading to the notion of “F1 geometry.’

Overview

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 23 / 75 Any idempotent algebra has “characteristic 1,” leading to the notion of “F1 geometry.’

Overview

A◦ = {a◦ : a ∈ A}.

(−)a is called the quasi-negative of a. A semigroup (A, +) has characteristic k > 0 if k + 1 = 1 with k ≥ 1 minimal. A has characteristic 0 if A does not have characteristic k for any k ≥ 1.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 23 / 75 Overview

A◦ = {a◦ : a ∈ A}.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 23 / 75 Tb is a monoid (resp. group) whenever T is.

Overview Symmetrized T -monoid modules

Although the max-plus algebra and its modules initially lack negation, one obtains negation maps of second kind for them through the next main idea, the symmetrization process, obtained by Gaubert (1992) in his dissertation, where an algebraic structure is embedded into a doubled structure with a natural negation map. Given any T -monoid module A, deﬁne Ab to be A(2) = A × A, with componentwise addition. Also deﬁne Tb = (T × {0}) ∪ ({0} × T ) with multiplication Tb × Ab → Ab given by

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

We also deﬁne a negation map given by the “switch”

(−)(a0, a1) = (a1, a0).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 24 / 75 Overview Symmetrized T -monoid modules

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

We also deﬁne a negation map given by the “switch”

(−)(a0, a1) = (a1, a0).

Tb is a monoid (resp. group) whenever T is. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 24 / 75 In particular, Nb is itself a semiring with negation given by (−)(m, n) = (n, m), which we call Z. The diﬀerence from the construction of Z from N, is that here we distinguish (m, n) from (m + k, n + k). When A has multiplication, Ab looks like a superalgebra, in the sense that one deﬁnes multiplication

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

Any congruence can be viewed naturally as a substructure of Ab. D. Joo and K. Mincheva have used this to good effect in defining prime congruences, and their definition generalizes to triples.

Overview

This is reminiscent of the familiar construction of Z from N, where (m, n) is identiﬁed with −(n, m).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 25 / 75 When A has multiplication, Ab looks like a superalgebra, in the sense that one deﬁnes multiplication

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

Any congruence can be viewed naturally as a substructure of Ab. D. Joo and K. Mincheva have used this to good effect in defining prime congruences, and their definition generalizes to triples.

Overview

This is reminiscent of the familiar construction of Z from N, where (m, n) is identiﬁed with −(n, m). In particular, Nb is itself a semiring with negation given by (−)(m, n) = (n, m), which we call Z. The diﬀerence from the construction of Z from N, is that here we distinguish (m, n) from (m + k, n + k).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 25 / 75 Any congruence can be viewed naturally as a substructure of Ab. D. Joo and K. Mincheva have used this to good effect in defining prime congruences, and their definition generalizes to triples.

Overview

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

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 25 / 75 Overview

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

Any congruence can be viewed naturally as a substructure of Ab. D. Joo and K. Mincheva have used this to good effect in defining prime congruences, and their definition generalizes to triples.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 25 / 75 Overview Modiﬁed symmetrized T -monoid modules

Here is an alternative version, due to Gaubert. Given any ordered monoid (G, ·), deﬁne Gb to be the subset of (G ∪ {0}) × (G ∪ {0}) generated by G × {0}, {0} × G and G × G, with componentwise multiplication and addition dominated by the larger component. For example, ( (a, a) if a ≥ b (a, a) + (b, 0) = (b, 0) if a < b.

Deﬁne T = G×, {0}, {0} × G, and (−)(a, 0) = (0, a), (−)(a, a) = (a, a).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 26 / 75 Overview Digression: Imposing distributivity

There is a cheap but useful way to give A a distributive multiplication, in cases where distributivity is lacking (as we shall see in some hyperﬁelds). Theorem: Any T -module A can be made (uniquely) into a semiring† via the multiplication !   X X X ai  bj  = ai bj . i j i,j

For the proof, it suﬃces to show that this is well-deﬁned, i.e., if P P 0 P P 0 0 i ai = i ai then i,j ai bj = i,j ai bj (and likewise for bj , bj ). But   ! ! X X X X X X X 0 X 0 ai bj =  ai bj  = ai bj = ai bj = ai bj . i,j i j j i j i i,j

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 27 / 75 ◦ We usually require that T ∩ A = ∅. (In particular 0 ∈/ T . We write T0 for T ∪ {0}.) (A, T , (−)) is called a T -group module triple when T is a multiplicative group. A triple (A, T , (−)) is a T -semiring triple if A is a semiring.

Overview Triples

(A, T , (−)) together (where (−) is a negation map with (−)T = T ) is called a pseudo-triple;(A, T , (−)) is a triple when T generates (A, +) additively.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 28 / 75 ◦ We usually require that T ∩ A = ∅. (In particular 0 ∈/ T . We write T0 for T ∪ {0}.) (A, T , (−)) is called a T -group module triple when T is a multiplicative group. A triple (A, T , (−)) is a T -semiring triple if A is a semiring.

Overview Triples

(A, T , (−)) together (where (−) is a negation map with (−)T = T ) is called a pseudo-triple;(A, T , (−)) is a triple when T generates (A, +) additively.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 28 / 75 (A, T , (−)) is called a T -group module triple when T is a multiplicative group. A triple (A, T , (−)) is a T -semiring triple if A is a semiring.

Overview Triples

(A, T , (−)) together (where (−) is a negation map with (−)T = T ) is called a pseudo-triple;(A, T , (−)) is a triple when T generates (A, +) additively.

◦ We usually require that T ∩ A = ∅. (In particular 0 ∈/ T . We write T0 for T ∪ {0}.)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 28 / 75 Overview Triples

(A, T , (−)) together (where (−) is a negation map with (−)T = T ) is called a pseudo-triple;(A, T , (−)) is a triple when T generates (A, +) additively.

◦ We usually require that T ∩ A = ∅. (In particular 0 ∈/ T . We write T0 for T ∪ {0}.) (A, T , (−)) is called a T -group module triple when T is a multiplicative group. A triple (A, T , (−)) is a T -semiring triple if A is a semiring.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 28 / 75 Unique negation fails in idempotent semirings in which negation is of the first kind, such as the max-plus, since any a ∈ T satisfies a = a + a = a◦ ∈ T ∩ A◦ = ∅. IMPORTANT: There is a big difference in taking a + b for a = (−)b, in which case it is a◦, and for a 6= (−)b. Accordingly, we need to exclude quasi-negatives from our criterion for bipotence.

Overview Uniquely negated triples

One of the key concepts: A triple (A, T , (−)) is uniquely negated if a + b ∈ A◦ for a, b ∈ T implies b = (−)a.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 29 / 75 Overview Uniquely negated triples

One of the key concepts: A triple (A, T , (−)) is uniquely negated if a + b ∈ A◦ for a, b ∈ T implies b = (−)a. Unique negation fails in idempotent semirings in which negation is of the first kind, such as the max-plus, since any a ∈ T satisfies a = a + a = a◦ ∈ T ∩ A◦ = ∅. IMPORTANT: There is a big difference in taking a + b for a = (−)b, in which case it is a◦, and for a 6= (−)b. Accordingly, we need to exclude quasi-negatives from our criterion for bipotence.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 29 / 75 The triples used in tropicalization (related to the max-plus algebra) are all (−)-bipotent, thereby motivating us to develop the algebraic theory of such triples. Any (−)-bipotent triple of the second kind is idempotent since (−)a 6= a implies a + a = max{a, a} = a. Conversely, any idempotent triple satisfying is of the second kind. The triple (Ab, Tb, (−)) is uniquely negated but not (−)-bipotent. The modiﬁed symmetrized T -monoid module is (−)-bipotent, which is why it is more useful at times. The following property, weaker than (−)-bipotence, actually is enough to carry the theory: A meta-tangible triple is a uniquely negated triple for which a + b ∈ T for any a 6= (−)b in T .

Overview Bipotent and meta-tangible triples

A triple (A, T , (−)) is (−)-bipotent if a + b ∈ {a, b} whenever a, b ∈ T with b 6= (−)a.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 30 / 75 The triple (Ab, Tb, (−)) is uniquely negated but not (−)-bipotent. The modiﬁed symmetrized T -monoid module is (−)-bipotent, which is why it is more useful at times. The following property, weaker than (−)-bipotence, actually is enough to carry the theory: A meta-tangible triple is a uniquely negated triple for which a + b ∈ T for any a 6= (−)b in T .

Overview Bipotent and meta-tangible triples

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 30 / 75 The following property, weaker than (−)-bipotence, actually is enough to carry the theory: A meta-tangible triple is a uniquely negated triple for which a + b ∈ T for any a 6= (−)b in T .

Overview Bipotent and meta-tangible triples

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 30 / 75 Overview Bipotent and meta-tangible triples

A triple (A, T , (−)) is (−)-bipotent if a + b ∈ {a, b} whenever a, b ∈ T with b 6= (−)a. The triples used in tropicalization (related to the max-plus algebra) are all (−)-bipotent, thereby motivating us to develop the algebraic theory of such triples. Any (−)-bipotent triple of the second kind is idempotent since (−)a 6= a implies a + a = max{a, a} = a. Conversely, any idempotent triple satisfying is of the second kind. The triple (Ab, Tb, (−)) is uniquely negated but not (−)-bipotent. The modiﬁed symmetrized T -monoid module is (−)-bipotent, which is why it is more useful at times. The following property, weaker than (−)-bipotence, actually is enough to carry the theory: A meta-tangible triple is a uniquely negated triple for which a + b ∈ T for any a 6= (−)b in T . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 30 / 75 The height of A is the maximal height of its elements. Thus A has height 2 iﬀ A = T0 ∪ (T + T ). Most systems arising in tropical mathematics have height 2, but height 3 provides new interesting examples.

Overview Height

We deﬁne the height of an element c ∈ A as the minimal t such that Pt c = i=1 ai with each ai ∈ T . (We say that 0 has height 0.)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 31 / 75 Overview Height

We deﬁne the height of an element c ∈ A as the minimal t such that Pt c = i=1 ai with each ai ∈ T . (We say that 0 has height 0.) The height of A is the maximal height of its elements. Thus A has height 2 iﬀ A = T0 ∪ (T + T ). Most systems arising in tropical mathematics have height 2, but height 3 provides new interesting examples.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 31 / 75 Overview Supertropical matrix theory – First Pass

Assume R = (R, G, ν) is a commutative supertropical domain†. One † deﬁnes the matrix semiring Mn(R) in the usual way. Since −1 is not available in tropical mathematics, we make do with the permanent, suggestively notated as |A|, and deﬁned for any matrix A = (ai,j ) as X |A| = aπ(1),1 ··· aπ(n),n. π∈Sn .

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 32 / 75 Theorem: |AB| = |A| |B| for n × n matrices over a supertropical semiring, whenever AB is nonsingular. 0 0 The assertion fails for AB nonsingular. For example, take A = . 1 2 1 2 |A| = 2, but A2 = , so A2 = 5ν 6= 4 = |A|2 . 3 4

0 0 Here A = is nonsingular, whereas A2 is singular. 1 2

Overview

Deﬁnition: An n × n matrix A is singular if |A| is tangible; A is singular when |A| ∈ G0.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 33 / 75 0 0 The assertion fails for AB nonsingular. For example, take A = . 1 2 1 2 |A| = 2, but A2 = , so A2 = 5ν 6= 4 = |A|2 . 3 4

0 0 Here A = is nonsingular, whereas A2 is singular. 1 2

Overview

Deﬁnition: An n × n matrix A is singular if |A| is tangible; A is singular when |A| ∈ G0.

Theorem: |AB| = |A| |B| for n × n matrices over a supertropical semiring, whenever AB is nonsingular.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 33 / 75 Overview

Deﬁnition: An n × n matrix A is singular if |A| is tangible; A is singular when |A| ∈ G0.

Theorem: |AB| = |A| |B| for n × n matrices over a supertropical semiring, whenever AB is nonsingular. 0 0 The assertion fails for AB nonsingular. For example, take A = . 1 2 1 2 |A| = 2, but A2 = , so A2 = 5ν 6= 4 = |A|2 . 3 4

0 0 Here A = is nonsingular, whereas A2 is singular. 1 2

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 33 / 75 Overview

Deﬁnition: We say b ghost surpasses a, written b |= a, if b = a + cν for some c. G

The correct theorem: Theorem: For any n × n matrices over a supertropical semiring R, we have

|AB| |= |A| |B| . G

In particular, |AB| = |A| |B| whenever AB is singular.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 34 / 75 Overview The surpassing relation

The last theorem suggests that we want to generalize equality on T to a relation on A which is not symmetric! Deﬁnition: A surpassing relation on A, denoted , is a partial pre-order satisfying the following, for elements of A:

1 0 a. ◦ ◦ 2 a b whenever a + c = b for some c ∈ A . 3 If a b then (−)a (−)b.

4 If ai bi for i = 1, 2 then a1 + a2 b1 + b2. 5 If a b for a, b ∈ T , then a = b. ◦ 6 a 6 b for any b ∈ T . A surpassing PO on A is a surpassing relation that restricts to a PO on A◦. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 35 / 75 One other property that one often wants is that a a◦, which holds in all of the tropical examples except the layered (when e0 6= e), but fails miserably in the classical case. Let us see why the conditions of the deﬁnition of surpassing relation are desired for to parallel equality.

(2) shows that reﬁnes ◦, and shows how the quasi-zeros behave like 0 under . (3), (4) are needed for considerations in universal algebra. (5) enables us to view as equality for tangible elements. (6) underlines the dichotomy between tangible elements and quasi-zeros.

Overview

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 meta-tangible triple.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 36 / 75 Overview

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 meta-tangible triple. Let us see why the conditions of the deﬁnition of surpassing relation are desired for to parallel equality.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 36 / 75 Here is a convenient way for building up triples and systems, based on our previous construction. Given triples (A`, T`, (−)) for ` ∈ L we form their (L) direct sum ⊕`∈L A`. This has been denoted A when each A` = A.

There are several natural options for T⊕A` , which should be clear according to the context, for c` ∈ A`: 1 T⊕A` = T , with the diagonal action a(c`) = (ac`) for a ∈ T . (This is useful in linear algebra, since we want to view T as scalars. This provides a quasi-triple but not a triple since it does not generate ⊕A`.) 2 T⊕A` = ∪T`. The action is defined componentwise, i.e., ak (c`) = ak ck for a` ∈ T`. The negation map also is defined componentwise. 3 P Same as in (2), but now T⊕A` = ` T` (which is generated by ∪T`). The action is defined componentwise, i.e., (a`)(c`) = (a`c`) for a` ∈ T`.

Overview Systems

A system (A, T , (−), ) is a uniquely negated triple (A, T , (−)) together with a T -surpassing relation , which often is a PO.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 37 / 75 1 T⊕A` = T , with the diagonal action a(c`) = (ac`) for a ∈ T . (This is useful in linear algebra, since we want to view T as scalars. This provides a quasi-triple but not a triple since it does not generate ⊕A`.) 2 T⊕A` = ∪T`. The action is defined componentwise, i.e., ak (c`) = ak ck for a` ∈ T`. The negation map also is defined componentwise. 3 P Same as in (2), but now T⊕A` = ` T` (which is generated by ∪T`). The action is defined componentwise, i.e., (a`)(c`) = (a`c`) for a` ∈ T`.

Overview Systems

A system (A, T , (−), ) is a uniquely negated triple (A, T , (−)) together with a T -surpassing relation , which often is a PO. Here is a convenient way for building up triples and systems, based on our previous construction. Given triples (A`, T`, (−)) for ` ∈ L we form their (L) direct sum ⊕`∈L A`. This has been denoted A when each A` = A.

There are several natural options for T⊕A` , which should be clear according to the context, for c` ∈ A`:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 37 / 75 Overview Systems

One can pretty well characterize the meta-tangible systems and recover the main examples in tropical mathematics, as well as some major examples in hyperﬁelds, to be discussed:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 38 / 75 0 (−) is of the ﬁrst kind. A = ∪m∈N mT , and e = 3. 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). 3 = 1. Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but one also has other examples. (−) is of the second kind. There are two possibilities: 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. T is not (−)-bipotent. Then the system is “classical.” Furthermore 3 = 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.

Overview

Theorem: Any metatangible group module system (A, T , (−), ) must satisfy one of the following:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 39 / 75 (−) is of the second kind. There are two possibilities: 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. T is not (−)-bipotent. Then the system is “classical.” Furthermore 3 = 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.

Overview

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 39 / 75 Overview

Theorem: Any metatangible group module system (A, T , (−), ) must satisfy one of the following: 0 (−) is of the ﬁrst kind. A = ∪m∈N mT , and e = 3. 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). 3 = 1. Hence (A, T , −, ) has characteristic 2. One example is the classical algebra of characteristic 2, but one also has other examples. (−) is of the second kind. There are two possibilities: 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. T is not (−)-bipotent. Then the system is “classical.” Furthermore 3 = 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.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 39 / 75 Overview Examples of systems

First the T -Semiring† and T -semifield† systems that we discussed. Classical algebra was considered above. Here the quasi-negative is the ◦ usual negative, which is unique, and A = {0}. a ◦ b iff b = a + 0 = a, so we have the T -system (A, T , −, =), which is meta-tangible. The negation map is of second kind unless A has characteristic 2, in which case (−) is of the first kind. This helps to “explain” why the theory of meta-tangible T -systems of first kind often has the flavor of characteristic 2. In the max-plus algebra the quasi-negatives are far from unique, since whenever b < a we have a + b = a = a◦. Height 2. These provide tropical structures designed to refine the max-plus algebra. All of them are (−)-bipotent T -systems, to be studied in depth. The familiar examples have characteristic 0, although some constructions can also be replicated in positive characteristic. Louis Rowen,Supertropical Bar-Ilan University semiringsA general† and algebraic the structure “symmetrized” theory for tropical mathematicsTTuesday-system 20 June, were 2017 40 / 75 considered above. Supertropical semirings† are precisely the semiring systems of the first kind of height 2, of characteristic 0. The negation map is given by (−)(k, a) = (−k, a). Thus the quasi-zeros will be of level 1 − 1.

Overview Layered semirings†

“Layered semirings” are 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 fact, associativity of multiplication in G is irrelevant, so we will call them “layered semialgebras.”. Addition is given by:  (` , a ) if a > a ;  1 1 1 2 (`1, a1) + (`2, a2) = (`2, a2) if a1 < a2; .  (`1 + `2, a1) if a1 = a2.

T = {±1} × G. 1A = (1, 1) ∈ T , and by induction, for k ∈ N, k = (k, 1) = (k − 1, 1) + (k, 1) = 1 + ··· + 1, taken k times. The (k, 1) generate a sub-semiring with negation map, and A = ∪k∈L(k, 1)T .

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 41 / 75 Overview Layered semirings†

T = {±1} × G. 1A = (1, 1) ∈ T , and by induction, for k ∈ N, k = (k, 1) = (k − 1, 1) + (k, 1) = 1 + ··· + 1, taken k times. The (k, 1) generate a sub-semiring with negation map, and A = ∪k∈L(k, 1)T .The negation map is given by (−)(k, a) = (−k, a). Thus the quasi-zeros will be of level 1 − 1. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 41 / 75 L = Z with the usual negation, T = {(`, a) ∈ L × G : ` = ±1}, and (−)(`, a) = (−`, a), of the second kind. This is useful for tropical integration. L is the residue ring of a valuation, where now T = {(`, a) ∈ L × G : ` 6= 0}. L is a ﬁnite ﬁeld of characteristic 2, where T = {(`, a) ∈ L × G : ` 6= 0}, and (−) is the identity.

Overview

Here are some natural explicit examples of layered semialgebras:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 42 / 75 L is the residue ring of a valuation, where now T = {(`, a) ∈ L × G : ` 6= 0}. L is a ﬁnite ﬁeld of characteristic 2, where T = {(`, a) ∈ L × G : ` 6= 0}, and (−) is the identity.

Overview

Here are some natural explicit examples of layered semialgebras:

L = N, formally with −` = `, T = {(`, a) ∈ L × G : ` = 1}, and (−) is the identity (thus of the ﬁrst kind). T ◦ is the layer 2. (The higher levels, if they exist, are neither tangible nor in T ◦. In fact e0 = 1 + 1 + 1 has layer 3.) This is useful for tropical diﬀerentiation. It has height equal to the cardinality of the submonoid of L generated by 1. It often provides counterexamples to assertions that hold in height 2. L = Z with the usual negation, T = {(`, a) ∈ L × G : ` = ±1}, and (−)(`, a) = (−`, a), of the second kind. This is useful for tropical integration.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 42 / 75 L is a ﬁnite ﬁeld of characteristic 2, where T = {(`, a) ∈ L × G : ` 6= 0}, and (−) is the identity.

Overview

Here are some natural explicit examples of layered semialgebras:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 42 / 75 Overview

Here are some natural explicit examples of layered semialgebras:

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 42 / 75 (The truncated algebra) A weird example, which leads to counterexamples in linear algebra and must be confronted. Fixing n > 1, we say that L = {1,..., n} is truncated at n if 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. The negation map is the identity. This T -triple has characteristic 0, since m 6= 1 for all m > 1, but it has height n. L is some classical algebraic structure, such as a ring, or an exterior algebra, or a Lie algebra.

Overview

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 43 / 75 L is some classical algebraic structure, such as a ring, or an exterior algebra, or a Lie algebra.

Overview

A somewhat more esoteric example from the tropical standpoint, but quite signiﬁcant algebraically. Fixing n > 0, taking L = Zn, identify each level modulo n. (This has height n and characteristic n.) (The truncated algebra) A weird example, which leads to counterexamples in linear algebra and must be confronted. Fixing n > 1, we say that L = {1,..., n} is truncated at n if 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. The negation map is the identity. This T -triple has characteristic 0, since m 6= 1 for all m > 1, but it has height n.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 43 / 75 Overview

k1 + k2 = n in L if k1 + k2 ≥ n in N;

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 43 / 75 Overview

The “exploded” T -system, where A = L × G with L the set of lowest coeﬃcients of Puiseux series, T = (L \ 0) × G, and (−)(`, a) = (−`, a), is (−)-bipotent of the second kind, provided L is not of characteristic 2.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 44 / 75 Overview Hypergroups

Recent interest has arisen in the study of hypergroups and hyperfields. It turns out that the hypergroups can be injected naturally into their power sets, which have a negation map, whereby the hyperfield is identified with the subset of singletons. The idea is to formulate all of our extra structure in terms of addition (and possibly other operations such as multiplication) on P(T ), the set of subsets of T , viewed as an additive semigroup, identifying T with the singletons in P(T ). But this is not so easy since T0 itself need not be closed under addition.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 45 / 75 Overview

The “intuitive” deﬁnition: A hyper-semigroup should be a structure (T , , 0) where : T × T → P(T ), for which the analog of associativity holds: (a1 a2) a3 = a1 (a2 a3), ∀a ∈ T .

There is a fundamental difficulty in this definition: a1 a2 need not be a singleton, so technically (a1 a2) a3 is not defined. This difficulty is exacerbated when considering generalized associativity; for example, what does (a1 a2) (a3 a4) mean in general?

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 46 / 75 We always think of in terms of addition. Note that repeated addition in the hyper-semigroup need not be deﬁned until one passes to the power set, which makes it diﬃcult to check basic universal relations such as associativity.

Overview

A hyper-semigroup is (T0, , 0), where is a commutative binary operation T0 × T0 → P(T0), 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 T0. 0 is the neutral element.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 47 / 75 Overview

A hyper-semigroup is (T0, , 0), where is a commutative binary operation T0 × T0 → P(T0), 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 T0. 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 the power set, which makes it diﬃcult to check basic universal relations such as associativity.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 47 / 75 A hypergroup is a hyper-semigroup (T , , 0) for which every element a has a unique hypernegative −a. The hypernegation is a negation map, and induces a negation map on P(T0), via (−)S = {−s : s ∈ S}. A T -hyperzero of a hypergroup (T , , 0) is a set of the form a (−a) ∈ P(T ). (This is not the usual definition, which is any subset of T containing 0, but serves just as well since, by definition, if 0 ∈ a b for a, b ∈ T then b = −a, implying a b is a hyperzero in our sense.) (T , , ·, 1) is a hyperfield if (T , , 0) is also a group (T , ·, 1).

Overview

A hypernegative of an element a in a hyper-semigroup (T , , 0) is an element −a for which 0 ∈ a (−a).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 48 / 75 The hypernegation is a negation map, and induces a negation map on P(T0), via (−)S = {−s : s ∈ S}. A T -hyperzero of a hypergroup (T , , 0) is a set of the form a (−a) ∈ P(T ). (This is not the usual definition, which is any subset of T containing 0, but serves just as well since, by definition, if 0 ∈ a b for a, b ∈ T then b = −a, implying a b is a hyperzero in our sense.) (T , , ·, 1) is a hyperfield if (T , , 0) is also a group (T , ·, 1).

Overview

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 48 / 75 A T -hyperzero of a hypergroup (T , , 0) is a set of the form a (−a) ∈ P(T ). (This is not the usual definition, which is any subset of T containing 0, but serves just as well since, by definition, if 0 ∈ a b for a, b ∈ T then b = −a, implying a b is a hyperzero in our sense.) (T , , ·, 1) is a hyperfield if (T , , 0) is also a group (T , ·, 1).

Overview

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 48 / 75 (T , , ·, 1) is a hyperﬁeld if (T , , 0) is also a group (T , ·, 1).

Overview

A hypernegative of an element a in a hyper-semigroup (T , , 0) is an element −a for which 0 ∈ a (−a). A hypergroup is a hyper-semigroup (T , , 0) for which every element a has a unique hypernegative −a. The hypernegation is a negation map, and induces a negation map on P(T0), via (−)S = {−s : s ∈ S}. A T -hyperzero of a hypergroup (T , , 0) is a set of the form a (−a) ∈ P(T ). (This is not the usual deﬁnition, which is any subset of T containing 0, but serves just as well since, by deﬁnition, if 0 ∈ a b for a, b ∈ T then b = −a, implying a b is a hyperzero in our sense.)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 48 / 75 Overview

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 48 / 75 Overview Here are some main examples of hyperﬁelds.

The tropical hyperfield. Define R∞ = R ∪ {−∞} and define the product a J 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 (satisfying Property P), easily seen to be isomorphic (as semirings) to Izhakian’s extended tropical arithmetic, where we identify (−∞, a] := {c : c ≤ a} with aν, and have a natural hyperfield isomorphism of this tropical hyperfield with the sub-semiring Rd∞ of P(R∞), because  b if b > a;  (−∞, a] + b = (−∞, a] if b = a (−∞, b] ∪ (b, a] = (−∞, a] if b < a.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 49 / 75 (Hyperfield of signs) Let S := {0, 1, −1} with the usual multiplication law and hyperaddition defined by 1 1 = {1}, −1 −1 = {−1}, x 0 = 0 x = {x}, and 1 −1 = −1 1 = {0, 1, −1} = S. Then S is a hyperfield (satisfying Property P), called the hyperfield of signs. The four elements {{0}, {−1}, {1}, S} constitute the sub-semiring† Se of P(S), and comprises a meta-tangible system.

Overview

The Krasner hyperﬁeld. 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 of the monoid K, where we identify {0; 1} with 1ν.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 50 / 75 Overview

(Hyperfield of signs) Let S := {0, 1, −1} with the usual multiplication law and hyperaddition defined by 1 1 = {1}, −1 −1 = {−1}, x 0 = 0 x = {x}, and 1 −1 = −1 1 = {0, 1, −1} = S. Then S is a hyperfield (satisfying Property P), called the hyperfield of signs. The four elements {{0}, {−1}, {1}, S} constitute the sub-semiring† Se of P(S), and comprises a meta-tangible system.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 50 / 75 Overview

The phase hyperﬁeld. Take T = S1, the complex unit circle, together with the center {0}, and Points a and b are antipodes if a = −b. Multiplication is deﬁned as usual (so corresponds on S1 to addition of angles). We call an arc of less than 180 degrees short.  all points in the short arc from a to b if a 6= b;  a b = {−a, 0, a} if a = −b 6= 0; {a} if b = 0.

T is a hyperfield, called the phase hyperfield. At the power set level, 1 given W1, W2 ⊆ S , we define W1 W2 to be the union of all (short) arcs from a point of W1 to a non-antipodal point in W2 (which together makes a connected arc), together with {0} if W2 contains an antipode of W1.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 51 / 75 Overview

Thus the system spanned by T is not meta-tangible, and its elements can be described as follows: {0}, which has height 0, T , the points on S1, which has height 1, Short arcs (the sum of non-antipodal distinct points), which have height 2, The sets {a, 0, −a} = a − a, which we write as a◦, which have height 2, Semicircles with 0, having the form a◦ + b where b 6= ±a, which have height 3, S1 ∪ {0} = a◦ + b◦ where b 6= ±a.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 52 / 75 Overview

+ Viro’s “triangle” hyperﬁeld T deﬁned over R by the formula

+ a b = {c ∈ R : |a − b| ≤ c ≤ a + b}.

In other words, c ∈ a b iﬀ there exists a Euclidean triangle with sides of lengths a, b, and c. Here

T + T = {[a1, a2]: a1 ≤ a2},

although not meta-tangible, has height 2, since a1+a2 a2−a1 ˆ 0 0 [a1, a2] = 2 + 2 ∈ A whereas [a1, a2] + [a1, a2] is some interval 0 going up to a2 + a2.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 53 / 75 Bipotent hypergroup systems include the “tropical hyperfield,” the Krasner hypergroup (of the first kind), and the sign hypergroup (of the second kind), all of height 2. The phase hypergroup system is idempotent of height 3, but not meta-tangible. The “triangle” hyperfield system is of the first kind and not idempotent. Here distributivity holds only with respect to elements of T , although this can be rectified by means of our earlier digression.

Overview Hypergroup systems

(P(T ), T , (−), ⊆) is a system, which we call a hypergroup system.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 54 / 75 Overview Hypergroup systems

(P(T ), T , (−), ⊆) is a system, which we call a hypergroup system. Bipotent hypergroup systems include the “tropical hyperfield,” the Krasner hypergroup (of the first kind), and the sign hypergroup (of the second kind), all of height 2. The phase hypergroup system is idempotent of height 3, but not meta-tangible. The “triangle” hyperfield system is of the first kind and not idempotent. Here distributivity holds only with respect to elements of T , although this can be rectified by means of our earlier digression.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 54 / 75 Any strongly negated T -system with respect to ◦ satisﬁes the fuzzy property.

Overview

Fuzzy rings can also be put into the context of systems. Deﬁnition: The fuzzy property for a triple (A, T , (−)) is: 0 ◦ 0 ◦ 0 0 ◦ 0 a1(−)a1 ∈ A and a2(−)a2 ∈ A imply a1a2(−)a1a2 ∈ A , ∀ai , ai ∈ A. This matches with Dress’ deﬁnition of a fuzzy ring.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 55 / 75 Overview

Any strongly negated T -system with respect to ◦ satisﬁes the fuzzy property.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 55 / 75 Overview Module triples over “fundamental” triples

Given a triple (A, T , (−)), which we call “fundamental,” we can deﬁne a (A, T , (−))-module triple (M, T (M), (−)) where M acts on A and T (M) acts on T . This leads to a representation theory of systems.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 56 / 75 We deﬁne T (M ⊗ N ) to be the simple tensors a ⊗ b where a ∈ T (M) and b ∈ T (N ).

Overview Tensor products

The tensor product is a very well studied construction in category theory, which can be done in universal algebra. We do this for systems (M, T (M), (−)) and (N , T (N), (−)), incorporating the negation map into the tensor product, deﬁning a negation map on M ⊗C N by (−)(v ⊗ w) = ((−)v) ⊗ w. We deﬁne a negated tensor product by imposing the extra axiom

((−)v) ⊗ w = v ⊗ ((−)w).

(This is done by modding out by the congruence generated by all elements ((−)v ⊗ w, v ⊗ (−)w) to the congruence deﬁning the tensor product in the universal algebra framework.)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 57 / 75 Overview Tensor products

((−)v) ⊗ w = v ⊗ ((−)w).

(This is done by modding out by the congruence generated by all elements ((−)v ⊗ w, v ⊗ (−)w) to the congruence deﬁning the tensor product in the universal algebra framework.)

We deﬁne T (M ⊗ N ) to be the simple tensors a ⊗ b where a ∈ T (M) and b ∈ T (N ).

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 57 / 75 Overview

Although they lose (−)-bipotence, tensor products provide a powerful tool in the theory.

One can construct the tensor system P M⊗i , which, as in classical i∈N theory, yields a host of structures including polynomial triples and others to be discussed.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 58 / 75 Overview Morphisms

Morphisms are deﬁned in terms of the surpassing relation. Deﬁnition: 0 0 0 Let A := (A1,... At ) and A := (A1,... At ) be carriers (of the same signature) with abstract surpassing relations and 0, respectively. A 0 0 -morphism f : A → A is a set of maps fj : Aj → Aj , 1 ≤ j ≤ t, satisfying the properties:

1 0 f (ω(a1,..., am)) ω(f (a1),..., f (am)), ∀aj ∈ Aji . 2 If aj bj in Aji for each j, then

0 f (ω(a1,..., am)) ω(f (b1),..., f (bm)).

As a special case we have: A morphism f : A → A0 of systems is a morphism, in particular satisfying f (TA) ⊂ TA0 .

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 59 / 75 Let us apply the tropicalization of the previous section to obtain tropical analogs of classical algebraic structures that are not necessarily commutative or even associative. We focus on two major instances — Exterior semialgebras and Lie semialgebras.

Overview Tropical structures arising from tropicalization

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 60 / 75 Overview Tropical structures arising from tropicalization

A key philosophical point: Tropicalization can be viewed as a morphism from the classical Puiseux series system to a tropical system. This provides a cookbook for deﬁning tropical structures in terms of systems, such as tropical Grassmann semialgebras, super-semialgebras, Lie semialgebras, and Poisson semialgebras. Let us apply the tropicalization of the previous section to obtain tropical analogs of classical algebraic structures that are not necessarily commutative or even associative. We focus on two major instances — Exterior semialgebras and Lie semialgebras.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 60 / 75 Overview Exterior (Grassmann) semialgebras with a negation map

As in the classical case, for free modules, the tensor semialgebra yields a construction of the Grassmann semialgebra whose base is the union of even elements and odd elements. Deﬁnition: A (faithful) Grassmann, or exterior, semialgebra, over a C-module V with a negation map, is a semialgebra A generated by V , together with a negation map extending (−) and a product A × A → A satisfying

1 v 2 ∈ A◦ for v ∈ V ,

v1v2 = (−)v2v1 for vi ∈ V ,

(−)(v1 ··· vt ) = ((−)v1)v2 ··· vt .

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 61 / 75 Overview

sgn(π) Thus vπ(1) ··· vπ(t) = (−) v1 ··· vt . The appropriate T -triple is (A, T , (−)), where T = {v1 ··· vt : vi ∈ T , t ∈ N}, the submonoid generated by V . v1v2 = (−)v2v1 is central in A, for all v1, v2 ∈ V .

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 62 / 75 Overview

Given a Grassmann semialgebra G over a module V with a negation map (−), we deﬁne G0 to be the submodule of G generated by all even products of elements of V , and G1 to be the submodule of G generated by all odd products of elements of V . Lemma

G = G0 + G1. G0 is in the center of G, and G1 = G0V . When V is the free module with negation, then G = G0 ⊕ G1 is a superalgebra.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 63 / 75 Overview Super-semialgebras

As in the classical case, one can “superize” the various classes in universal algebra, mimicking the standard classical way of making a theory super. Deﬁnition:

The Grassmann envelope of a super-semialgebra A = A0 ⊕ A1 is the sub-semialgebra (A0 ⊗ G0) ⊕ (A1 ⊗ G1) of A ⊗ G, with G as in Lemma 1. (Thus we view the Grassmann envelope without the grading.) Suppose V is a variety of universal algebras. A super-V semialgebra is a super-semialgebra A whose Grassmann envelope is in V.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 64 / 75 Overview

For example, ij A is super-commutative if ai aj = (−) aj ai whenever ai ∈ Ai , aj ∈ Aj . ij+1 A is super-anticommutative if ai aj = (−) aj ai whenever ai ∈ Ai , aj ∈ Aj . The Grassmann envelope of the Grassmann super-semialgebra G itself is (G0 ⊗ G0) ⊕ (G1 ⊗ G1) which is commutative, so G is “super-commutative.”

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 65 / 75 Overview Lie semialgebras and Lie super-semialgebras, and their triples

We turn again to tropicalization for the tropical version of Lie algebras. A semialgebra A with negation map is anticommutative if it satisﬁes the conditions for all a, b ∈ A: 2 ◦ 1 a ∈ A ; 2 ba = (−)(ab) = a((−)b) = ((−)a)b.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 66 / 75 Lemma

ad is a morphism from L to EndF L. (In fact ad preserves addition.) Furthermore, [[ab]v] [a[bv]](−)[b[av]] for all a, b, v ∈ L.

Proof. Follows from the deﬁnitions.

This can be viewed as the ◦-surpassing version of Jacobi’s identity.

Overview Deﬁnition: A Lie semialgebra with a negation map (over a semiring F ) is a module L with a negation map (−), endowed with anticommutative multiplication L × L → L, written (a, b) 7→ [ab], called a Lie bracket (in view of the standard notation [ab] for Lie multiplication), satisfying

ad[ab] [ada, adb] ∀a, b ∈ L. (3)

(Note that we do not require a negation map on F .)

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 67 / 75 Overview Deﬁnition: A Lie semialgebra with a negation map (over a semiring F ) is a module L with a negation map (−), endowed with anticommutative multiplication L × L → L, written (a, b) 7→ [ab], called a Lie bracket (in view of the standard notation [ab] for Lie multiplication), satisfying

ad[ab] [ada, adb] ∀a, b ∈ L. (3)

(Note that we do not require a negation map on F .)

Lemma ad is a morphism from L to EndF L. (In fact ad preserves addition.) Furthermore, [[ab]v] [a[bv]](−)[b[av]] for all a, b, v ∈ L.

Proof. Follows from the deﬁnitions.

This can be viewed as the ◦-surpassing version of Jacobi’s identity. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 67 / 75 Overview Matrices over systems

Matrices over a uniquely negated system yield a system. Indeed, for 2 (n) m = n , as with classical algebra, Mn(A) ≈ EndA(A ) has the module structure of A(n2) for any semiring A, and we get a system, taking T (Mn(A)) = Mn(T ), and deﬁning (−) and componentwise, (Mn(T ), ·) need no longer be a monoid even when (T , ·) is a monoid. Nevertheless, signiﬁcant results are available, coupling standard computations with the transfer principle.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 68 / 75 Overview Determinants over a T -monoid module with a negation map

Suppose A has a negation map (−). For a permutation π, write ( a : π even; (−)πa = (−)a : π odd.

The (−)-determinant |A| of a matrix A is ! X π Y (−) ai,π(i) . π∈Sn i

The even part is P Q a , and the odd part is π∈Sn even i i,π(i) P Q a . π∈Sn odd i i,π(i) A matrix A is singular if |A| ∈ A◦. A is nonsingular if |A| ∈/ A◦.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 69 / 75 Overview Dependence relations of vectors

A vector v ∈ M is called tangible if each of its entries is in T0. Thus, a matrix is tangible iﬀ each of its rows is tangible. Suppose that M is an A-module. A set S ⊆ M is T -dependent if there are v1,..., vm ∈ S and (nonzero) αj ∈ T such that m X ◦ αj vj ∈ M . j=1 Otherwise S is T -independent.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 70 / 75 Overview

An element v ∈ M is T -dependent on a T -independent set S ⊆ M, written a ∈dep S, if S ∪ {a} is T -dependent. An element v ∈ M is strongly T -dependent on a T -independent set S ⊆ M, written v ∈dep S, if there are v1,..., vm ∈ S and (nonzero) αj ∈ T such that m X v αj vj . j=1

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 71 / 75 Overview Ranks of matrices

Our next task is to compare diﬀerent notions of rank of matrices, in terms of its row vectors and its column vectors. We only consider tangible matrices A, i.e., with entries in T0, for meta-tangible systems. Deﬁnition The (surpassing) row rank of a matrix A is the maximal number of T -independent rows of A. The column rank of the matrix A is the maximal number of T -independent columns of A. The submatrix rank of the matrix A is the maximal k such that A has a nonsingular k × k submatrix.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 72 / 75 Overview

Let is consider the following assertions:

1 Condition A1: The submatrix rank is less than or equal to the row rank and the column rank. 2 Condition A2: The three deﬁnitions of rank are equal for any tangible matrix, when T is a multiplicative group. An easy induction argument enables one to reduce Condition A1 to proving that a square matrix A is singular if its rows are dependent, which is our next result. Theorem: If the rows of a tangible n × n matrix A over a cancellative meta-tangible triple are dependent, then |A| ∈ A◦.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 73 / 75 Overview

The converse holds for supertropical algebras, but there is a counterexample when (−) is of the second kind, due to Akian, Gaubert, and Guterman: Let T := {0, 1, −1} be the hyperﬁeld of signs. Let us write + for +1 and − for −1. The matrix + + − + + − + + (4) − + + + has row rank 3, but every square submatrix is singular.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 74 / 75 Thank you for your attention, and a happy non-birthday to Eric.

Overview Closing philosophy

Every tropical version of an algebraic structure should be a system. Then tropicalization is a morphism from the classical system of the Puiseux series algebra to a meta-tangible system.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 75 / 75 Overview Closing philosophy

Every tropical version of an algebraic structure should be a system. Then tropicalization is a morphism from the classical system of the Puiseux series algebra to a meta-tangible system.

Thank you for your attention, and a happy non-birthday to Eric.

Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 75 / 75