Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the \emph{prime $k$-spectrum} of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the \emph{space of valuations} and the \emph{space of prime congruences} on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of \emph{closure operations} for semirings, and provide several examples. In particular, we introduce an \emph{integral closure operation} and a \emph{Frobenius closure operation} for idempotent semirings.
Skew idempotent functionals of ordered semirings are studied. Different associative and non-associative semirings are considered. Theorems about properties of skew idempotent functionals are proved. Examples are given.
Since for the classification of finite (congruence-)simple semirings it remains to classify the additively idempotent semirings, we progress on the characterization of finite simple additively idempotent semirings as semirings of join-morphisms of a semilattice. We succeed in doing this for many cases, amongst others for every semiring of this kind with an additively neutral element. As a consequence we complete the classification of finite simple semirings with an additively neutral element. To complete the classification of all finite simple semirings it remains to classify some very specifi…
Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well known semiring. The construction of an equivalence relation such that any equivalence class contain just one idempotent is proposed. In our main result we prove that such equivalence class is a semiring and find his order. We prove that…
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.
Here we characterize the linear operators that preserve rank of matrices over additively idempotent and multiplicatively cancellative semirings. The main results in this article generalize the corresponding results on the two element Boolean algebra and on the max algebra; and holds on max-plus algebra and some other tropical semirings.
The resultant plays a crucial role in (computational) algebra and algebraic geometry. One of the most important and well known properties of the resultant is that it is equal to the determinant of the Sylvester matrix. In 2008, Odagiri proved that a similar property holds over the tropical semiring if one replaces subtraction with addition. The tropical semiring belongs to a large family of algebraic structures called commutative idempotent semiring. In this paper, we prove that the same property (with subtraction replaced with addition) holds over an \emph{arbitrary\/} commutative idempotent …
We prove that the variety V of commutative multiplicatively idempotent semirings satisfying x + y + xyz = x + y is generated by single semirings. Moreover, we describe a normal form system for terms in V and we show that the word problem in V is solvable. Although V is locally finite, it is residually big.
Here we describe the least distributive lattice congruence $η$ on an idempotent semiring in general and characterize the varieties $D^\bullet, L^\bullet$ and $R^\bullet$ of all idempotent semirings such that $η=\mathcal{D}^\bullet, \mathcal{L}^\bullet$ and $\mathcal{R}^\bullet$, respectively. If $S \in D^\bullet [L^\bullet, R^\bullet]$, then the multiplicative reduct $(S, \cdot)$ is a [left, right] normal band. Every semiring $S \in D^\bullet$ is a spined product of a semiring in $L^\bullet$ and a semiring in $R^\bullet$ with respect to a distributive lattice.
Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.
Let $R$ be a commutative additively idempotent semiring. In this paper, some properties and characterizations for permanents of matrices over $R$ are established, and several inequalities for permanents are given. Also, the adjiont matrices of matriecs over $R$ are considered. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices.
Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras
Acharya Nagarjuna University（阿查里亚·纳加尔朱纳大学）；Government College (Autonomous), Rajahmundry（拉贾穆恩德里政府学院（自治））；Government College for Women (Autonomous), Guntur（贡图尔女子政府学院（自治））
We investigate endomorphism semirings of a finite semilattice with one least element and one greatest element such that all the other elements form an antichain. We construct some new finite simple semirings. Keywords: endomorphism semiring, additively idempotent semiring, simple semiring, idempotent endomorphism, finite semilattice.
In our previous work, motivated by the study of tropical polynomials, a definition for prime congruences was given for an arbitrary commutative semiring. It was shown that for additively idempotent semirings this class exhibits some analogous properties to prime ideals in ring theory. The current paper focuses on the resulting notion of Krull dimension, which is defined as the length of the longest chain of prime congruences. Our main result states that for any additively idempotent semiring $A$, the semiring of polynomials $A[x]$ and the semiring of Laurent polynomials $A(x)$, we have $\dim A…
In this talk a brief survey of basic ideas of Idempotent Mathematics is presented. Relations between this theory and the theory of fuzzy sets as well as the possibility theory and some applications (including computer applications) are discussed.
We explore Jordan derivations of triangular matrices with entries from an additively idempotent semiring. The main result states that for any matrix A over additively idempotent semiring, if we put all the elements of the family of dense submatrices of A to be zeroes, we find a derivative of A. The set of derivations of this type is established.
In this note we construct a solution of a matrix interval linear equation of the form X=AX+B (the discrete stationary Bellman equation) over partially ordered semirings, including the semiring of nonnegative real numbers and all idempotent semirings. We discuss also the computational complexity of problems in interval idempotent linear algebra. In the traditional Interval Analysis problems of this kind are generally NP-hard. In the note we consider matrix equations over positive semirings; in this case the computational complexity of the problem is polynomial. Idempotent and other positive sem…
It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is o…
In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. We present several results addressed toward a semiring theory for MV-algebras. In particular we show a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, the construction of the Grothendieck group of a semiring and its functorial nature, and the effect of Mundici categorical eq…
We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.
We show that, if S is a finite semiring, then the free profinite S-semimodule on a Boolean Stone space X is isomorphic to the algebra of all S-valued measures on X, which are finitely additive maps from the Boolean algebra of clopens of X to S. These algebras naturally appear in the logic approach to formal languages as well as in idempotent analysis. Whenever S is a (pro)finite idempotent semiring, the S-valued measures are all given uniquely by continuous density functions. This generalises the classical representation of the Vietoris hyperspace of a Boolean Stone space in terms of continuou…
Using the closure operator that defines a subtractive ideal of a semiring $S$, in this note we introduce a topology on the set of all ideals of $S$ induced by that operator. We show that the corresponding subtractive space is $T_0$ and every nonempty irreducible closed set has a unique generic point, whereas the restricted subspace of subtractive ideals is $T_1$. Using a semiring homomorphism, we obtain a continuous map between the corresponding subtractive spaces.
There is a well known correspondence between the triangle inequality for a distance function on a finite set, and idempotency of an associated matrix over the tropical semiring. Recent research has shed new light on the structure (algebraic, combinatorial and geometric) of tropical idempotents, and in this paper we explore the consequences of this for the metric geometry of tropical polytopes. We prove, for example, that every n-point metric space is realised by the Hilbert projective metric on the vertices of a pure n-dimensional tropical polytope in tropical n-space. More generally, every n-…
Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings
Department of Mathematics, Government College (Autonomous), Y-Junction, Rajahmundry, 533105, A.P.,India（政府学院（自治），Y交汇处，拉贾穆恩德里）；Department of Mathematics, Acharya Nagarjuna University, Pedakakani, Guntur, 522510, A.P., India（阿查里亚·纳加尔琼纳大学）；Department of Mathematics, Government College For Women (A), Pattabhipuram, Guntur, 522006, A.P., India（女子政府学院（A））
We develop an extension of valuations theorem for suitable extensions of idempotent semirings. As an application, we give a new proof for the classical case of fields. Along the way, we develop characteristic one analogues of some central results in the theory of valuation rings.
An incline is an additively idempotent semiring in which the product of two elements is always less than or equal to either factor. This paper proves that the only regular inclines are distributive lattices, which also implies that there is no noncommutative regular incline.
We present a theory of lattice-enriched semirings, called quantic semirings, which generalize both quantales and powersets of hyperrings. Using these structures, we show how to recover the spectrum of a Krasner hyperring (and in particular, a commutative ring with unity) via universal constructions, and generalize the spectrum to a new class of hyperstructures, hypersemirings. (These include hyperstructures currently studied under the name "semihyperrings", but we have weakened the distributivity axioms.) Much of the work consists of background material on closure systems, suplattices, quantal…
An incline is an additively idempotent semiring in which the product of two elements is always less than or equal to either factor. By making use of prime numbers, this paper proves that A^{11} is less than or equal to A^5 for all 3x3 matrices A over an arbitrary commutative incline, thus giving an answer to an open problem "For 3x3 matrices over any incline (even noncommutative) is X^5 greater than or equal to X^{11}?", proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984.
