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The category $\operatorname{STROP}$ of commutative semirings, whose morphisms are transmissions, is a full and reflective subcategory of the category $\operatorname{STROP}_m$ of supertropical monoids. Equivalence relations on supertropical monoids are constructed easily, and utilized effectively for supertropical semirings, whereas ideals are too special for semirings. Aiming for tangible factorizations, certain types of such equivalence relations are constructed and classified explicitly in this paper, followed by a profound study of their characteristic properties with special emphasis on difficulties arising from ghost products of tangible elements.

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We complement two papers on supertropical valuation theory ([1111.
Izhakian , Z. ,
Knebusch , M. ,
Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 . Preprint at arXiv:1003.1101 .[CrossRef], [Web of Science ®]View all references], [1212.
Izhakian , Z. ,
Knebusch , M. ,
Rowen , L. ( 2013 ). Dominance and transmissions in supertropical valuation theory . Commun. Algebra 41 ( 7 ): 2736 – 2782 .[Taylor & Francis Online], [Web of Science ®]View all references]) by providing natural examples of m-valuations (= monoid valuations), and afterwards of supervaluations and transmissions between them. These supervaluations have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. We develop the basics of the theory of such semirings and transmissions.

This paper is a sequel of [IKR1], where we defined supervaluations on a
commutative ring $R$ and studied a dominance relation $\phi \geq \psi$ between
supervaluations $\phi$ and $\psi$ on $R$, aiming at an enrichment of the
algebraic tool box for use in tropical geometry.
A supervaluation $\phi:R \to U$ is a multiplicative map from $R$ to a
supertropical semiring $U$, cf. [IR1], [IR2], [IKR1], with further properties,
which mean that $\phi$ is a sort of refinement, or covering, of an m-valuation
(= monoid valuation) $v: R \to M$. In the most important case, that $R$ is a
ring, m-valuations constitute a mild generalization of valuations in the sense
of Bourbaki [B], while $\phi \geq \psi$ means that $\psi: R \to V$ is a sort of
coarsening of the supervaluation $\phi$. If $\phi(R)$ generates the semiring
$U$, then $\phi \geq \psi$ iff there exists a "transmission" $\alpha: U \to V$
with $\psi = \alpha \circ \phi$.
Transmissions are multiplicative maps with further properties, cf. [IKR1,
Sec. 5]. Every semiring homomorphism $\alpha: U \to V$ is a transmission, but
there are others which lack additivity, and this causes a major difficulty. In
the main body of the paper we study surjective transmissions via equivalence
relations on supertropical semirings, often much more complicated than
congruences by ideals in usual commutative algebra.

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 a d-base may vary according to the d-base, 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. Linear functionals and the dual space are also studied, leading to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence, as well as a supertropical version of a theorem of Artin. Comment: 28 pages

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. Comment: 16 pages

Supertropical monoids are a structure slightly more general than the
supertropical semirings, which have been introduced and used by the first and
the third authors for refinements of tropical geometry and matrix theory in
[IR1]-[IR3], and then studied by us in a systematic way in [IKR1]-[IKR3] in
connection with "supervaluations".
In the present paper we establish a category $\STROP_m$ of supertropical
monoids by choosing as morphisms the "transmissions", defined in the same way
as done in [IKR1] for supertropical semirings. The previously investigated
category $STROP$ of supertropical semirings is a full subcategory of $STROP_m.$
Moreover, there is associated to every supertropical monoid $V$ a supertropical
semiring $\hat V$ in a canonical way.
A central problem in [IKR1]-[IKR3] has been to find for a supertropical
semiring $U$ the quotient $U/E$ by a "TE-relation", which is a certain kind of
equivalence relation on the set $U$ compatible with multiplication (cf. [IK1,
Definition 4.5]). It turns out that this quotient always exists in $\STROP_m$.
In the good case, that $U/E$ is a supertropical semiring, this is also the
right quotient in $\STROP.$ Otherwise, analyzing $(U/E)^\wedge,$ we obtain a
mild modification of $E$ to a TE-relation $E'$ such that $U/E' = (U/E)^\wedge$
in $\STROP.$
In this way we now can solve various problems left open in [IKR1], [IKR2] and
gain further insight into the structure of transmissions and supervaluations.
Via supertropical monoids we also obtain new results on totally ordered
supervaluations and monotone transmissions studied in [IKR3].

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.

We interpret a valuation v on a ring R as a map v:R→M into a so-called bipotent semiring M (the usual max–plus setting), and then define a supervaluationφ as a suitable map into a supertropical semiring U with ghost ideal M (cf. Izhakian and Rowen (2010, in press) and ) covering v via the ghost map U→M. The set Cov(v) of all supervaluations covering v has a natural ordering which makes it a complete lattice. In the case where R is a field, and hence for v a Krull valuation, we give a completely 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.

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 although unique factorization may fail, there is a “preferred” factorization that is explained both geometrically and algebraically. The failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.

This paper, a continuation of Izhakian and Rowen (in press) [5], involves a closer study of polynomials over supertropical semirings and their version of tropical geometry. We introduce the concept of relatively prime polynomials (in one indeterminate) and resultants, with the aid of some topology. Polynomials in one indeterminant are seen to be relatively prime iff they do not have a common tangible root, iff their resultant is tangible. Applying various morphisms of supertropical varieties leads to a supertropical version of Bézout's theorem.

We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a generalized eigenspace decomposition of a power of an arbitrary supertropical matrix. Comment: 21 pages

This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular. Comment: 17 pages, 2 figures

EpU, xq. Notations 7.15. (a) If z P T pUq, c :" ez, we call an S c pzq-path also an Ispzq-path. Thus, if z is a son of x P T pUq, z " px, an elementary Ispzq

- P U Puq

Case I: rxs Sispxq is tangible. Then all sons of rxs Sispxq in U{Sispxq are isolated.
Case II: rxs Sispxq is ghost. Then Sispxq " EpU, xq.
Notations 7.15.
(a) If z P T pUq, c :" ez, we call an S c pzq-path also an Ispzq-path. Thus, if z is a
son of x P T pUq, z " px, an elementary Ispzq-path is a triple pvpx, u, wpxq with
u, v, w, p P U, vpx P T pUq, wpx P T pUq, and evpx " ewpx " epx.

we call an Spxq-path (cf. Definition 6.12) also an Sispxq-path. Thus, an elementary Sispxq-path is an elementary Ispzq-path for some son z

- If X P T Puq

If x P T pUq, we call an Spxq-path (cf. Definition 6.12) also an Sispxq-path. Thus,
an elementary Sispxq-path is an elementary Ispzq-path for some son z " px of x.

- Z Izhakian

Z. Izhakian. Commutative ν-algebra and supertropical algebraic geometry, arXiv:1901.08032, 2019.