Jens Zumbrägel
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.
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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.
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.
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…
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…
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。
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…
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…
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.
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 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.
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.
On the Jacobson radical and semisimplicity of a semiring
基于极小与单表示，本文引入半环的两个 Jacobson 型 Hoehnke 根，将其表示为正则同余的交，并刻画半单半环与 s-本原半环的结构。
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…
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…
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…
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。
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.
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。
The finite basis problem for matrix semirings over a two-element additively idempotent semiring
研究了二维加法幂等半环上矩阵半环的有限基性质，证明当半环不同于分配格时，矩阵半环具有有限基。
Structures of $(m,n)$-seminearring
该研究借鉴$(m,n)$-半环理论，引入并系统建立$(m,n)$-半近环及其理想、同态、商结构等基本概念与性质。
The twisted partition monoid $\mathcal{P}_n^Φ$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main result of this paper is a complete description of the congruences on $\mathcal{P}_n^Φ$. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of $n+1$ congruences on the additive monoid $\mathbb{N}$ of natural numbers and a certain $(n+1)\times\mathbb{N}$ matrix. We also give a description of the inclusion ordering of congruences in t…
Computational and Categorical Frameworks of Finite Ternary $Γ$-Semirings: Foundations, Algorithms, and Industrial Modeling Applications
本文提出基于约束的枚举算法，研究有限三元Γ半环的结构理论，实现分类与范畴合成，支持工业建模应用。
We classify essential algebras whose irredundant non-refinable covers consist of primal algebras. The proof is obtained by constructing one to one correspondence between such algebras and partial orders on finite sets. Further, we prove that for a finite algebra, it has an irredundant non-refinable cover consists of primal algebras if and only if it is the both congruence primal and arithmetical. Finally, we obtain combinatorial description of congruence primal arithmetical algebras.
Finite diagonalizable torsors and Kummer cohomology over semiring schemes
研究对秩一正则半模\(R\)上\(R\)-三线性交换\(\Gamma\)-索引三元积分类，其积具特定形式，等同于带标记系数交换半环。由此得函子性基变换和泛局部化定理，还在弱假设下识别相关元素、构建谱等，通过实例区分假设展示局部行为。
On generators of commutative semifields
本文研究理想单交换半环的分类，并针对（准）半域证明其最小生成元数与相关根森林深度呈线性关系。
$F$-congruences on $F$-inverse monoids
该研究刻画了F-逆幺半群上F-同余对应的核正规系与同余对，还完成了F-无同余的F-逆幺半群的分类，丰富了半群代数理论中F-逆幺半群的同余结构研究。
Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups
本文研究有限全变换半群变体中极大正则子半群的同余格，通过将其分解为 $Cong(T_r)$ 与组合系统的等价关系格的次直积，给出同余的显式分类及格的高的公式。
We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep(O(\infty) (formally), Rep(O(N), Rep(Sp(N) or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. Note that the paper is followed by a brief erratum, wh…
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