An informal overview of triples and systems
---
paper_title: Algebraic Geometry Over Hyperrings

paper_content:

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is ‘multi-valued’. This paper largely consists of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme (X,O X ) ( X , O X ) and prove that Γ(X,O X )≃R Γ ( X , O X ) ≃ R for any integral affine hyperring scheme X = Spec R .
---
paper_title: Supertropical matrix algebra

paper_content:

The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. * There exists an adjoint matrix $\adj{A}$ such that the matrix $A \adj{A}$ behaves much like the identity matrix (times $|A|$). * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix. * The tropical determinant of a matrix $A$ is a ghost iff the rows of $A$ are tropically dependent, iff the columns of $A$ are tropically dependent. * Every root of $f_A$ is a"supertropical"eigenvalue of $A$ (appropriately defined), and has a tangible supertropical eigenvector.
---
paper_title: Algebras with a negation map

paper_content:

Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras, one often is challenged by the lack of negation. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called \textbf{systems}, in the context of universal algebra, showing how these unify the more viable (super)tropical versions, as well as hypergroup theory and fuzzy rings, thereby "explaining" similarities in their theories. Special attention is paid to \textbf{meta-tangible} $\mathcal T$-systems, whose algebraic theory includes all the main tropical examples and many others, but is rich enough to facilitate computations and provide a host of structural results. ::: Basic results also are obtained in linear algebra, linking determinants to linear independence. ::: Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby further explaining the mysterious link between classical algebraic results and their tropical analogs, as well as with hyperfields. We utilize the tropicalization functor to propose tropical analogs of classical algebraic notions. ::: The systems studied here might be called "fundamental," since they are the underlying structure which can be studied via other "module" systems, which is to be the third stage of this project, involving a theory of sheaves and schemes and derived categories with a negation map.
---
paper_title: The geometry of blueprints. Part I: Algebraic background and scheme theory

paper_content:

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\ congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as"improved"cyclotomic field extensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and"archimedean valuation rings". It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.
---
paper_title: Tropical Arithmetic and Matrix Algebra

paper_content:

This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.
---
paper_title: Matroids over hyperfields

paper_content:

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give"cryptomorphic"axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.
---
paper_title: Algebraic structures of tropical mathematics

paper_content:

Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid $\tM$ is replaced by $R: = L\times \tM$, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When $L=\{1,\infty\}$ we recover the"standard"supertropical structure with its"ghost"layer. When $L = \NN $ we can describe multiple roots of polynomials via a"layering function"$s: R \to L$. Likewise, one can define the layering $s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination $\sum \a_i v_i$ is a ghost, for"tangible"$\a_i \in R$. Then an $n\times n$ matrix has tropically dependent rows iff its permanent is a ghost. We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraic foundation for tropical linear algebra, in which many classical tools are available. In the process, we provide some new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework of identities of semirings.
---
paper_title: Supertropical algebra

paper_content:

We develop the algebraic polynomial theory for"supertropical algebra,"as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of"ghost elements,"which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz. Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and unique factorization holds in a certain sense. On the other hand, the failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.
---
paper_title: Categories with negation

paper_content:

We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is prime, and there is a weak Nullstellensatz. Also, the polynomial $\mathcal A[\la_1, \dots, \la_n]$ and Laurent polynomial systems $\mathcal A[[\la_1, \dots, \la_n]]$ in $n$ commuting indeterminates over a $\tT$-semiring-group system have dimension $n$. For module systems, special attention also is paid to tensor products and $\Hom$. Abelian categories are replaced by"semi-abelian"categories (where $\Hom(A,B)$ is not a group) with a negation morphism.
---
paper_title: On the relation between hyperrings and fuzzy rings

paper_content:

We construct a full embedding of the category of hyperfields into Dress’s category of fuzzy rings and explicitly characterize the essential image—it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker and Bowler’s theory of strong matroids over hyperfields with Dress’s theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker and Bowler’s partial demifields.
---
paper_title: Projective module systems

paper_content:

We develop the basic theory of projective modules and splitting in the more general setting of systems. This enables us to prove analogues of classical theorems for tropical and hyperfield theory. In this context we prove a Dual Basis Lemma and develop Morita theory. We also prove a Schanuel's Lemma as a first step towards defining homological dimension.
---
paper_title: Duality theory for finite and infinite matroids with coefficients

paper_content:

On presente une theorie de la dualite pour les matroides finis et infinis avec coefficients
---
paper_title: Homological algebra in characteristic one

paper_content:

This article develops several main results for a general theory of homological algebra in categories such as the category of idempotent modules. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category $\bmod$ of modules over the Boolean semifield $\B:=\{0,1\}$ is the replacement for the category of abelian groups. We show that the semi-additive category $\bmod$ fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on $\bmod$ we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category $\b2$ is simply $\bmod$ in the topos of sets endowed with an involution and as such it shares with $\bmod$ most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category $\b2$ is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between $\b2$ and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of $\b2$, the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
---
paper_title: Algebras with a negation map

paper_content:

Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras, one often is challenged by the lack of negation. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called \textbf{systems}, in the context of universal algebra, showing how these unify the more viable (super)tropical versions, as well as hypergroup theory and fuzzy rings, thereby "explaining" similarities in their theories. Special attention is paid to \textbf{meta-tangible} $\mathcal T$-systems, whose algebraic theory includes all the main tropical examples and many others, but is rich enough to facilitate computations and provide a host of structural results. ::: Basic results also are obtained in linear algebra, linking determinants to linear independence. ::: Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby further explaining the mysterious link between classical algebraic results and their tropical analogs, as well as with hyperfields. We utilize the tropicalization functor to propose tropical analogs of classical algebraic notions. ::: The systems studied here might be called "fundamental," since they are the underlying structure which can be studied via other "module" systems, which is to be the third stage of this project, involving a theory of sheaves and schemes and derived categories with a negation map.
---
paper_title: The geometry of blueprints. Part I: Algebraic background and scheme theory

paper_content:

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\ congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as"improved"cyclotomic field extensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and"archimedean valuation rings". It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.
---
paper_title: Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3

paper_content:

We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree K 2 ≥ 3, where in the case of surfaces of degree 3 with two real components we introduce a certain modification of Welschinger invariants and enumerate exclusively the curves traced on the non-orientable component. As an application, we prove the positivity of the invariants under consideration and their logarithmic asymptotic equivalence, as well as congruence modulo 4, to genus zero Gromov–Witten invariants.
---
paper_title: Tropicalization of Del Pezzo Surfaces

paper_content:

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schl\"afli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces.
---
paper_title: Methods and Applications of (max,+) Linear Algebra

paper_content:

Exotic semirings such as the “(max, +) semiring” (ℝ ∪ {−∞},max,+), or the “tropical semiring” (ℕ ∪ {+∞}, min, +), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities).
---
paper_title: Combinatorial and inductive methods for the tropical maximal rank conjecture

paper_content:

Abstract We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m = 3 , extending previous results for m = 2 .
---
paper_title: Max-algebra: the linear algebra of combinatorics?

paper_content:

Abstract Let a ⊕ b =max( a , b ), a ⊗ b = a + b for a,b∈ R := R ∪{−∞} . By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue–eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra.
---
paper_title: Introduction to Tropical Geometry

paper_content:

Tropical islands Building blocks Tropical varieties Tropical rain forest Tropical garden Toric connections Bibliography Index
---
paper_title: Homological algebra in characteristic one

paper_content:

This article develops several main results for a general theory of homological algebra in categories such as the category of idempotent modules. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category $\bmod$ of modules over the Boolean semifield $\B:=\{0,1\}$ is the replacement for the category of abelian groups. We show that the semi-additive category $\bmod$ fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on $\bmod$ we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category $\b2$ is simply $\bmod$ in the topos of sets endowed with an involution and as such it shares with $\bmod$ most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category $\b2$ is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between $\b2$ and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of $\b2$, the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
---
paper_title: Linear independence over tropical semirings and beyond

paper_content:

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.
---
paper_title: Tropical Cramer Determinants Revisited

paper_content:

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.
---
paper_title: Supertropical matrix algebra

paper_content:

The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. * There exists an adjoint matrix $\adj{A}$ such that the matrix $A \adj{A}$ behaves much like the identity matrix (times $|A|$). * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix. * The tropical determinant of a matrix $A$ is a ghost iff the rows of $A$ are tropically dependent, iff the columns of $A$ are tropically dependent. * Every root of $f_A$ is a"supertropical"eigenvalue of $A$ (appropriately defined), and has a tangible supertropical eigenvector.
---
paper_title: Supertropical semirings and supervaluations

paper_content:

We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a \textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U \to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $v$ a Krull valuation, we give a complete explicit description of $\Cov(v)$. ::: The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.
---
paper_title: Tropical Arithmetic and Matrix Algebra

paper_content:

This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.
---
paper_title: Supertropical algebra

paper_content:

We develop the algebraic polynomial theory for"supertropical algebra,"as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of"ghost elements,"which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz. Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and unique factorization holds in a certain sense. On the other hand, the failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.
---
paper_title: Layered Tropical Mathematics

paper_content:

Abstract Generalizing supertropical algebras, we present a “layered” structure, “sorted” by a semiring which permits varying ghost layers, and indicate how it is more amenable than the “standard” supertropical construction in factorizations of polynomials, description of varieties, and for mathematical analysis and calculus, in particular with respect to multiple roots of polynomials. This gives rise to a significantly better understanding of the tropical resultant and discriminant. Explicit examples and comparisons are given for various sorting semirings such as the natural numbers and the positive rational numbers, and we see how this theory relates to some recent developments in the tropical literature.
---
paper_title: Supertropical Matrix Algebra II: Solving tropical equations

paper_content:

We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of $A$, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to $A$; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer's rule. We also describe various properties of this tangible adjoint, and use it to compute supertropical eigenvectors, thereby producing an example in which an $n\times n$ matrix has $n$ distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.
---
paper_title: Categories of layered semirings

paper_content:

We generalize the constructions of [17,19] to layered semirings, in order to enrich the structure and provide finite examples for applications in arithmetic (including finite examples). The layered category theory of [19] is extended accordingly, to cover noncancellative monoids.
---
paper_title: Valuations of Semirings

paper_content:

We develop a notion of valuations on a semiring. In particular, we classify valuations on the semifield $\mathbb{Q}_{max}$ and also valuations on the (suitably defined) `function field' $\mathbb{Q}_{max}(T)$ which are trivial on $\mathbb{Q}_{max}$. As a byproduct, we reinterpret the projective line $\mathbb{P}^1_{\mathbb{F}_1}$ over $\mathbb{F}_{1}$ as an abstract curve associated to $\mathbb{Q}_{max}(T)$.
---
paper_title: Algebraic Geometry Over Hyperrings

paper_content:

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is ‘multi-valued’. This paper largely consists of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme (X,O X ) ( X , O X ) and prove that Γ(X,O X )≃R Γ ( X , O X ) ≃ R for any integral affine hyperring scheme X = Spec R .
---
paper_title: Matroids over hyperfields

paper_content:

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give"cryptomorphic"axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.
---
paper_title: Tropical schemes, tropical cycles, and valuated matroids

paper_content:

The tropicalization of a subvariety Y in the n-dimensional algebraic torus T is a polyhedral complex trop(Y ) that is a \combinatorial shadow" of the original variety. Some invariants of Y , such as the dimension, are encoded in trop(Y ). The complex trop(Y ) comes equipped with positive integer weights on its top-dimensional cells, called multiplicities, that make it into a tropical cycle. This extra information encodes ::: information about the intersection theory of compacti cations of the original variety Y ; see for example [KP11].
---
paper_title: Matroids over hyperfields

paper_content:

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give"cryptomorphic"axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.
---
paper_title: On the relation between hyperrings and fuzzy rings

paper_content:

We construct a full embedding of the category of hyperfields into Dress’s category of fuzzy rings and explicitly characterize the essential image—it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker and Bowler’s theory of strong matroids over hyperfields with Dress’s theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker and Bowler’s partial demifields.
---
paper_title: Duality theory for finite and infinite matroids with coefficients

paper_content:

On presente une theorie de la dualite pour les matroides finis et infinis avec coefficients
---
paper_title: Linear independence over tropical semirings and beyond

paper_content:

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.
---
paper_title: Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials

paper_content:

A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield ${\pmb T}$, the semifield $\mathbb{Z}_{max}$ and the two element semifield $\mathbb{B}$. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension $1$. An improvement of a result of A. Bertram and R. Easton from 2013 is proven which can be regarded as a Nullstellensatz for tropical polynomials.
---
paper_title: Methods and Applications of (max,+) Linear Algebra

paper_content:

Exotic semirings such as the “(max, +) semiring” (ℝ ∪ {−∞},max,+), or the “tropical semiring” (ℕ ∪ {+∞}, min, +), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities).
---
paper_title: Tropical Cramer Determinants Revisited

paper_content:

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.
---
paper_title: Categories with negation

paper_content:

We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is prime, and there is a weak Nullstellensatz. Also, the polynomial $\mathcal A[\la_1, \dots, \la_n]$ and Laurent polynomial systems $\mathcal A[[\la_1, \dots, \la_n]]$ in $n$ commuting indeterminates over a $\tT$-semiring-group system have dimension $n$. For module systems, special attention also is paid to tensor products and $\Hom$. Abelian categories are replaced by"semi-abelian"categories (where $\Hom(A,B)$ is not a group) with a negation morphism.
---
paper_title: Homological algebra in characteristic one

paper_content:

This article develops several main results for a general theory of homological algebra in categories such as the category of idempotent modules. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category $\bmod$ of modules over the Boolean semifield $\B:=\{0,1\}$ is the replacement for the category of abelian groups. We show that the semi-additive category $\bmod$ fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on $\bmod$ we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category $\b2$ is simply $\bmod$ in the topos of sets endowed with an involution and as such it shares with $\bmod$ most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category $\b2$ is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between $\b2$ and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of $\b2$, the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
---
paper_title: Algebraic Geometry Over Hyperrings

paper_content:

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is ‘multi-valued’. This paper largely consists of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme (X,O X ) ( X , O X ) and prove that Γ(X,O X )≃R Γ ( X , O X ) ≃ R for any integral affine hyperring scheme X = Spec R .
---
paper_title: Algebras with a negation map

paper_content:

Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras, one often is challenged by the lack of negation. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called \textbf{systems}, in the context of universal algebra, showing how these unify the more viable (super)tropical versions, as well as hypergroup theory and fuzzy rings, thereby "explaining" similarities in their theories. Special attention is paid to \textbf{meta-tangible} $\mathcal T$-systems, whose algebraic theory includes all the main tropical examples and many others, but is rich enough to facilitate computations and provide a host of structural results. ::: Basic results also are obtained in linear algebra, linking determinants to linear independence. ::: Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby further explaining the mysterious link between classical algebraic results and their tropical analogs, as well as with hyperfields. We utilize the tropicalization functor to propose tropical analogs of classical algebraic notions. ::: The systems studied here might be called "fundamental," since they are the underlying structure which can be studied via other "module" systems, which is to be the third stage of this project, involving a theory of sheaves and schemes and derived categories with a negation map.
---
paper_title: Linear independence over tropical semirings and beyond

paper_content:

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.
---
paper_title: Tropical Cramer Determinants Revisited

paper_content:

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.
---
paper_title: Remarks on the Cayley-Hamilton Theorem

paper_content:

We revisit the classical theorem by Cayley and Hamilton,"{\em each endomorphism is a root of its own characteristic polynomial}", from the point of view of {\em Hasse--Schmidt derivations on an exterior algebra}
---
paper_title: Grassman semialgebras and the Cayley-Hamilton theorem

paper_content:

We develop a theory of Grassmann triples via Hasse-Schmidt derivations, which formally generalizes results such as the Cayley-Hamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.
---
paper_title: The geometry of blueprints. Part I: Algebraic background and scheme theory

paper_content:

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\ congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as"improved"cyclotomic field extensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and"archimedean valuation rings". It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.
---
paper_title: Categories with negation

paper_content:

We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is prime, and there is a weak Nullstellensatz. Also, the polynomial $\mathcal A[\la_1, \dots, \la_n]$ and Laurent polynomial systems $\mathcal A[[\la_1, \dots, \la_n]]$ in $n$ commuting indeterminates over a $\tT$-semiring-group system have dimension $n$. For module systems, special attention also is paid to tensor products and $\Hom$. Abelian categories are replaced by"semi-abelian"categories (where $\Hom(A,B)$ is not a group) with a negation morphism.
---
paper_title: Homological algebra in characteristic one

paper_content:

This article develops several main results for a general theory of homological algebra in categories such as the category of idempotent modules. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category $\bmod$ of modules over the Boolean semifield $\B:=\{0,1\}$ is the replacement for the category of abelian groups. We show that the semi-additive category $\bmod$ fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on $\bmod$ we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category $\b2$ is simply $\bmod$ in the topos of sets endowed with an involution and as such it shares with $\bmod$ most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category $\b2$ is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between $\b2$ and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of $\b2$, the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
---
paper_title: Introduction to Tropical Geometry

paper_content:

Tropical islands Building blocks Tropical varieties Tropical rain forest Tropical garden Toric connections Bibliography Index
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paper_title: Tropical schemes, tropical cycles, and valuated matroids

paper_content:

The tropicalization of a subvariety Y in the n-dimensional algebraic torus T is a polyhedral complex trop(Y ) that is a \combinatorial shadow" of the original variety. Some invariants of Y , such as the dimension, are encoded in trop(Y ). The complex trop(Y ) comes equipped with positive integer weights on its top-dimensional cells, called multiplicities, that make it into a tropical cycle. This extra information encodes ::: information about the intersection theory of compacti cations of the original variety Y ; see for example [KP11].
---
paper_title: Supertropical linear algebra

paper_content:

The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of “ghost surpasses”. Special attention is paid to the various notions of “base”, which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in various d-bases may differ, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identified with a set of “critical” elements. Then we turn to orthogonality of vectors, which leads to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence. We also obtain a supertropical version of a theorem of Artin, which says that if g-orthogonality is a symmetric relation, then the underlying bilinear form is (supertropically) symmetric.
---

Title: An Informal Overview of Triples and Systems
Section 1: Introduction
Description 1: Provide an overview of the goals, motivations, and the scope of the paper, including the unified theory of tropical algebra, hyperfields, and fuzzy rings.

Section 2: Acquaintance with Basic Notions
Description 2: Introduce the fundamental concepts and structures, including T-modules, quasi-zeros, and the importance of universal algebra in discussing diverse structures.

Section 3: Motivating Examples
Description 3: Elaborate on non-classical examples that motivate the theory, such as supertropical semirings, hyperfields, and fuzzy rings.

Section 4: Idempotent Semirings
Description 4: Discuss the role of idempotent semirings in tropical geometry and their key properties and applications.

Section 5: Supertropical Semirings
Description 5: Describe the structure and significance of supertropical semirings and how they relate to classical algebraic geometry and linear algebra.

Section 6: Hyperfields and Other Related Constructions
Description 6: Explore the concept of hyperfields and other algebraic constructions that replace elements with sets in sums.

Section 7: Fuzzy Rings
Description 7: Provide an overview of fuzzy rings and their relationship to hypergroups and matroids.

Section 8: Symmetrization
Description 8: Explain the symmetrization construction and its applications to algebra, including defining T-supermodules and switch maps.

Section 9: Negation Maps, Triples, and Systems
Description 9: Discuss the implementation of negation maps in semirings and their significance in the study of triples and systems.

Section 10: Ground Triples versus Module Triples
Description 10: Compare ground triples and module triples, highlighting their respective roles in structure theory and representation theory.

Section 11: Contents of [64]: Meta-Tangible Systems
Description 11: Summarize the key concepts and results from the paper [64], focusing on meta-tangible systems and their axioms.

Section 12: Contents of [4]: Linear Algebra over Systems
Description 12: Outline the main contributions of the paper [4] regarding linear algebra over semiring systems, including definitions of T-dependent vectors and matrix ranks.

Section 13: Contents of [20]: Grassmann Semialgebras
Description 13: Unify classical and tropical theory through the study of Grassmann semialgebras, including Hasse-Schmidt derivations and the generalization of the Cayley-Hamilton theorem.

Section 14: Contents of [50]: Basic Categorical Considerations
Description 14: Detail the categorical aspects of systems as discussed in [50], including functors, prime systems, and congruences.

Section 15: Contents of [48]: Projective Module Systems
Description 15: Discuss the definition and fundamental properties of projective module systems as elaborated in [48].

Section 16: Interface between Systems and Tropical Mathematics
Description 16: Relate systems to other approaches in tropical mathematics and describe the systemic version of tropical ideals.

Section 17: Areas for Further Research
Description 17: Identify areas for further research, including the study of affine varieties, valuated matroids, and their formulation over systems.