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The generalized L'vov-Kaplansky conjecture states that for any finite-dimensional simple algebra $A$ the image of a multilinear polynomial on $A$ is a vector space. In this paper we prove it for the algebra of octonions $\mathbb{O}$ over a field satisfying certain specified conditions (in particular, we prove it for quadratically closed field and for field $\mathbb{R}$). In fact, we prove that the image set must be either $\{0\}$, $F$, the space of pure octonions $V$, or $\mathbb{O}$. We discuss possible evaluations of semihomogeneous polynomials on $\mathbb{O}$ and of arbitrary polynomials on the corresponding Malcev algebra., 14 pages

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.

We find a semigroup [Formula: see text], whose category of partial representations contains the representation category [Formula: see text] of the free left regular band [Formula: see text]. We use this to construct a resolution for the absolute kernel of a representation of [Formula: see text], for which the kernel [Formula: see text] of the Markov operation in the Tsetlin library model is a prominent example. We obtain a formula for the dimension of the absolute kernel, generalizing the equality of the dimension of [Formula: see text] to the number of derangements of order [Formula: see text].

We introduce a theory of Clifford semialgebra systems, with application to representation theory via Hasse-Schmidt derivations on exterior semialgebras. Our main result, after the construction of the Clifford semialgebra, is a formula describing the exterior semialgebra as a representation of the Clifford semialgebra, given by the endomorphisms of the first wedge power., Comment: 43 pages

We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context., Comment: 23pp

An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0 . More generally, we call a submodule W of V “summand absorbing” (SA) in V if ∀ x , y ∈ V : x + y ∈ W ⇒ x ∈ W , y ∈ W . These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Holder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation.

Given a monic separable polynomial π of degree 2n over an arbitrary field and a scalar α, we define generic algebras Hπ and $\mathcal {A}_{\alpha \pi }$ for the decomposition of π into a product of two polynomials of degree n and for the factorization απ = 𝜃2 respectively. We investigate representations of degree 1 or 2 of these generic algebras. Every representation of degree 1 of Hπ factors through an etale algebra of degree ${2n}\choose n$ , whereas $\mathcal {A}_{\alpha \pi }$ has no representation of degree 1. We show that every representation of degree 2 of Hπ or $\mathcal {A}_{\alpha \pi }$ factors through the Clifford algebra of some quadratic form, pointed or not, and thus obtain a description of the quaternion algebras that are split by the etale algebra Fπ defined by π of by the function field of the hyperelliptic curve Xαπ with equation y2 = απ(x). We prove that every quaternion algebra split by the function field of Xαπ is also split by Fπ, and provide an example to show that a quaternion algebra split by Fπ may not be split by the function field of any curve Xαπ.

The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, withtheir introduction of {\it axial algebras}, and in particular {\it primitive axial algebras of Jordan type} (PJs for short). It turns out that these notions are closely related to $3$-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov, and M. Van Couwenberghe generalized axial algebrasto {\it decomposition algebras} which, in particular, are not necessarily commutative. This paper deals with decomposition algebras which are non-commutative versions of PJs., Comment: 12 pages

"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes of respective types $(\lambda,\delta)$ and $(\lambda',\delta')$ must be at most $5$; $d$ cannot be $4.$ If $d\le 3$ we list all the possibilities for $A$ up to isomorphism. We prove a variety of additional results and mention some research questions at the end., Comment: 16 pages

We develop the theory of l-weak identities in order to provide a feasible way of studying the central polynomials of matrix algebras. We describe the weak identities of minimal degree of matrix algebras in any dimension.

We describe triples and systems, expounded as an axiomatic algebraic umbrella theory for classical algebra, tropical algebra, hyperfields, and fuzzy rings.

The purpose of this note is to obtain precise information about associative or Jordan algebras generated by two idempotents.

For a maximal separable subfield K of a central simple algebra A, we provide a semiring isomorphism between K–K-sub-bimodules of A and H–H-sub-bisets of G = Gal ( L / F ) , where F = Cent ( A ) , L is the Galois closure of K / F , and H = Gal ( L / K ) . This leads to a combinatorial interpretation of the growth of dim K ⁡ ( ( K a K ) i ) , for fixed a ∈ A , especially in terms of Kummer subspaces.

In this paper, we develop the rudiments of a tropical homology theory. We use the language of “triples” and “systems” to simultaneously treat structures from various approaches to tropical mathematics, including semirings, hyperfields, and super tropical algebra. We enrich the algebraic structures with a negation map where it does not exist naturally. We obtain an analogue to Schanuel’s lemma which allows us to talk about projective dimension of modules in this setting. We define two different versions of homology and exactness, and study their properties. We also prove a weak Snake lemma type result.

We develop a theory of Grassmann semialgebra triples using 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.

We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular matrix $A$, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue $\lambda$, and corresponds to the columns of the eigenmatrix $A+\lambda I$ from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case a certain "difference criterion" holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix $A^\nabla : = \det(A)^{-1}\adj(A)$ and the connection of the independence question to generalized eigenvectors., Comment: The first author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowship

We classify the monomial Kummer subspaces of division symbol algebras of prime degree p, showing that every such space is standard, and in particular the dimension is no greater than p + 1 . It follows that in a generic symbol algebra, the dimension of any Kummer subspace is at most p + 1 .

This paper is a sequel to [6] , in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V → R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering , which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “ q -minimal elements,” which are elements a which cannot be written in the form b + c where b a but q ( b ) = q ( a ) . We determine the q -minimal elements by examining their support. But the class of all polynomial maps (in up to rank ( V ) variables) is itself a module over R , so the basic properties of the minimal ordering are applied to this R -module, or its submodule Quad ( V ) consisting of quadratic forms on V . This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics. Quad ( V ) is the sum of two disjoint submodules QL ( V ) and Rig ( V ) , consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6] ). Both QL ( V ) and Rig ( V ) are free with explicitly known bases, but Quad ( V ) itself is almost never free.

We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma., We changed the title and added more results

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set ${\rm sl}_n(K)$ of matrices of trace 0, or all of $M_n(K)$. This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for $n=2$ in Section 2, some decisive results for $n=3$ in Section 3, and partial information for $n\geq 3$ in Section 4, also for non-multilinear polynomials. In addition we consider the case of $K$ not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations., Comment: In this review, we present a systematized exposition including results obtained in our previous papers arXiv:1005.0191, arXiv:1306.4389, arXiv:1310.8563, arXiv:1310.1598, arXiv:1906.04973, arXiv:1506.06792. In addition, we have systematized ideas and methods of proofs

Any (associative) left Noetherian algebra over a field, which is finitely generated as an algebra over a central subring, is representable. As a special case, we give a short proof of the well-known theorem that any algebra over a field, which also is finite (as a module) over a central affine algebra, is representable. We give a counterexample to some natural generalizations, as well as a conjectured generic counterexample, together with specific positive results for irreducible algebras and finitely presented algebras. Also, based on joint work of the second author with Amitsur (posthumous), we show that any semiprimary PI-algebra with radical squared 0 is weakly representable.

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., 23 pages

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., 37 pages, extra material included to compare with other tropical approaches

The Zariski closure of an arbitrary representable (not necessarily associative) algebra is studied in the general context of universal algebra, with an application being that the codimension sequence is exponentially bounded.

A supertropical monoid is a monoid U together with a projection onto a totally ordered submonoid e U (where e ∈ U is idempotent). Supertropical monoids are slightly more general than the supertropical semirings that were introduced and used by the first and the third authors for refinements of tropical geometry and matrix theory, and then studied systematically by the authors in connection with “ supervaluations ”, and they permit a finer investigation of the supertropical theory. In the present paper we extend our earlier study of the category STROP of supertropical semirings to a category STROP m of supertropical monoids whose morphisms are “ transmissions ”, defined analogously as for supertropical semirings. Moreover, there is associated to every supertropical monoid V a canonical supertropical semiring V . A central problem in Izhakian et al. (2011) [8–10] has been to find the quotient U / E of a supertropical semiring U by a “ TE-relation ”, which is a certain kind of congruence. This quotient always exists in STROP m , and is the natural quotient in STROP in case U / E happens to be a supertropical semiring. Otherwise, analyzing ( U / E ) ∧ , we obtain a mild modification of E to a TE-relation E ′ such that U / E ′ = ( U / E ) ∧ in STROP . In this way we now can solve problems about universality in the category STROP that were left open in our earlier work, and gain further insight into the structure of transmissions and supervaluations which leads to new results on totally ordered supervaluations and monotone transmissions.

An [Formula: see text]-module [Formula: see text] over a semiring [Formula: see text] lacks zero sums (LZS) if [Formula: see text] implies [Formula: see text]. More generally, a submodule [Formula: see text] of [Formula: see text] is “summand absorbing” (SA), if, for all [Formula: see text], [Formula: see text] These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation.

In [7] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.

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 supertropical eigenspace decomposition of a power of an arbitrary supertropical matrix.

We construct, for any prime p p , a non-cyclic central simple algebra of degree p 2 p^2 with p 2 p^2 -central elements. This construction answers a problem of Peter Roquette.

The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows

We solve the following problem related to the Kneser–Tits conjecture, for Azumaya algebras. Given an Azumaya algebra D of rank 4 that is not a division algebra, whose center K is three-dimensional over the ground field F, such that cor K/F D is trivial, is it true that every element of D having reduced norm in F is a product of n elements having both reduced norm and reduced trace in F? This is true for n ≥ 3, but false for n = 2.

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, [Formula: see text]-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield[Formula: see text][Formula: see text], we pass to the semifield[Formula: see text][Formula: see text] of fractions of the polynomial semiring[Formula: see text], for which there already exists a well developed theory of kernels, which are normal convex subgroups of [Formula: see text]; the parallel of the zero set now is the [Formula: see text]-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to [Formula: see text]-kernels (Definition 4.1.4) and [Formula: see text]-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The [Formula: see text]-kernels corresponding to tropical hypersurfaces are the [Formula: see text]-sets of what we call “corner internal rational functions,” and we describe [Formula: see text]-kernels corresponding to “usual” tropical geometry as [Formula: see text]-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of [Formula: see text]-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between [Formula: see text]-sets and a class of [Formula: see text]-kernels of the rational [Formula: see text]-semifield[Formula: see text] called polars, originating from the theory of lattice-ordered groups. When [Formula: see text] is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal [Formula: see text]-kernels, intersected with the [Formula: see text]-kernel generated by [Formula: see text]. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of [Formula: see text]-kernels.

In this article, we develop further the theory of matrices over the extended tropical semiring. We introduce the notion of tropical linear dependence, enabling us to define matrix rank in a sense that coincides with the notions of tropical nonsingularity and invertibility.