Jawad Y. Abuhlail, Rangga Ganzar Noegraha
We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.
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We provide a classification of congruence-simple semirings with a multiplicatively absorbing element and without non-trivial nilpotent elements.
In this paper we investigate about the congruence generated by Y on quasi completely regular semirings and obtained the interval which Y* belongs to on quasi completely regular semirings.
For any semiring, the concept of k-congruences is introduced, criteria for k-congruences are established, it is proved that there is an inclusion-preserving bijection between k-congruences and k-ideals, and an equivalent condition for the existence of a zero is presented with the help of k-congruences. It is shown that a semiring is k-simple iff it is k-congruence-simple, and that inclines are k-simple iff they have at most 2 elements. Lemma 2.12(i) in [Glas. Mat. 42(62) (2007) 301] is pointed out being false.
Our main result states that a finite semiring of order >2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a `dense' subsemiring of the endomorphism semiring of a finite idempotent commutative monoid. We also investigate those subsemirings further, addressing e.g. the question of isomorphy.
In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set, and the second based on the layered structure. We tie them to tropical geometry, especially in connection with the dimension of an affine variety.
Finitely Generated Congruences in Semirings and Canonical Positive Models
本文研究所有同余有限生成的半环（称同余诺特半环），聚焦实序的典范正模型，证明其虽作为N-模非有限生成，但兼具同余诺特性与N上平坦性。
Congruence-simple multiplicatively idempotent semirings and multiplicative divisibility
该论文解决了关于乘法幂等半环的两个猜想，证明了同余单乘法幂等半环有限，以及有限生成交换乘法可除半环为乘法幂等，完善了半环相关分类理论。
In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zero-multiplication rings of prime order, matrix rings over finite fields, those that are additively idempotent, and those with trivial addition.
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…
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.
Congruence-simple matrix semirings
本文将斜域上全矩阵环为单环的经典定理推广到半环，刻画了矩阵半环为同余单的充分条件，即半环含乘法吸收元或为交换加法可消半环。
In this paper, among other results, there are described (complete) simple - simultaneously ideal- and congruence-simple - endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering th…
In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $SK_0(S)$ of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the $SK_0$-groups and $K_0$-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the $SK_0$-groups completely characterize zerosumfree congruence-semisimple semirings.
As pointed out in the monographs by J. S. Golan and by W. Kuich and A. Salomaa on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S1 and S2, one can ask whether an ideal I of their direct product S = S1 x S2 can be expressed in the form I1 x I2 where Ij is an ideal of Sj for j=1,2. Of course, the converse is elementary, namely if Ij is an ideal of Sj for j=1,2 then I1 x I2 is an ideal of S1 x S2. Having a congruence on a commutative semiring S, its 0-class is an ideal of S, but n…
On algebraic congruence varieties over semirings
本文为交换半环上的同余代数簇理论建立基础，证明了素同余谱的Zariski拓扑结构，并引入满足闭集公理的ρ-代数簇，得出同余版Nullstellensatz等结构结果。
In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and anti-bounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CI-semir…
Maximal-Hull $z$-Ideals, Congruence Closures, and Coherent Frames of Commutative Semirings
该研究为交换半环发展\(z\)-理想谱理论，证明\(z\)-理想格是凝聚框架，在特定假设下\(g\)-闭理想格也是，给出正则性准则，区分两种闭包并给出函子表述，扩展了环上相关定理。
The spectrum of prime congruences of a pair
本文将 Joo-Mincheva 的幂等半环素同余谱理论推广到半环对，证明其谱可嵌入半环对的素谱，并在正 $e$-型条件下成为满射。
On semisubtractive ideals of semirings
本文探讨了半环的半减法理想，证明了其构成完全模格，引入了 Golan 闭包及 $s$-强不可约与 $s$-不可约子类，研究了其与 $Q$-理想的关系、拓扑性质及 $s$-同余双射。
In this paper, we introduce homological structure theory of semirings and CP-semirings---semirings all of whose cyclic semimodules are projective. We completely describe semisimple, Gelfand, subtractive, and anti-bounded, CP-semirings. We give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CP-semirings, which solve two earlier open problems for these classes of semirings. We also study in detail the properties of semimodules over Boolean algebras whose endomorphism semirings are CP-semirings; and, as a consequence of this result, we give a comple…
The aim of this paper is to develop an `external' Kurosh-Amitsur radical theory of semirings and, using this approach, to obtain some fundamental results regarding two Jacobson type of radicals --- the Jacobson-Bourne, J-, radical and a very natural its variation, J_{s}-radical --- of hemirings, as well as the Brown-McCoy, R_{BM}-, radical of hemirings. Among the new central results of the paper, we single out the following ones: Theorems unifying two, internal and external, approches to the Kurosh-Amitzur radical theory of hemirings; A characterization of J-semisimple hemirings; A description…
In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, as well as establish some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a semifield S that extends the well-known characterizations when the ground semiring S is a field. Also, extending the well-known characterizations when S is a field or commutative ring, we present a complete characterization of congruence-simple Leavitt path algebras over row-fin…
This work obtains all the right ideals, radicals, congruences and ideals of the affine near-semirings over Brandt semigroups.
Typed Congruences and Quotient-Critical Irreducibility in Complete Ternary $Γ$-Semirings
该研究识别出三元Γ-乘积编码的交替五元核心，完成其范畴重构，建立二元坍缩与原子相关性质的等价关系，给出有限核心的相关算法，并构造无二元坍缩的指标敏感示例。
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…
We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the well-known characterizations when S is a field or a commutative ring. We also provide a criterion for the Steinberg algebra of the graph groupoid associated to an arbitrary graph to be congruence-simple. Motivated by a result of Clark and Sims, we show that, over the Boolean semifield, the natural homomorphism from the Leavitt path algebra to the Steinberg algebr…
Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings
本文研究了有限交换三元Γ半环的结构与极小理论，通过理想论方法建立了子直积分解定理，验证了理论的一致性。
Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum
本文研究了交换三元Γ半环中的素理想与半素理想，通过商结构和谱拓扑揭示了新的代数结构。
Concerning semirings of measurable functions
本研究探讨非负实值可测函数半环$\mathcal{M}^+(X,\mathcal{A})$的代数性质，证明其理想格与差环理想格同构、理想均为半环$z$-理想，研究可消同余与滤子的对偶性，给出紧可测空间新刻画，解决相关半环同构问题。
On the Subtractive Ideal Structure of Commutative Semirings
本文研究了交换半环中减法性质对理想结构的影响，揭示了k-素和k-半素理想的新特征化，并证明了相关理想类别的重合性。
From subtractive ideals of semirings to deductive and inductive sets in general algebras
本文将半环中核的特征化扩展到一般代数，并分析了不同代数设置中"减法"和"理想"的对应物。
The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of Bruhat--Tits trees. We also include some discussion and experimentation related to the congruence subgroup problem, which is open when $S$ contains at least two odd primes. In the appendix, we provi…
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