Semiring and semimodule issues in MV-algebras

Exact Sequences of Semimodules over Semirings

In this paper, we introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties.

Injective Semimodules - Revisited

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreove…

Fundamental Theorem of Projective Geometry over Semirings

We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry

Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules

In algebraic geometry over a variety of universal algebras $Θ$, the group $Aut(Θ^{0})$ of automorphisms of the category $Θ^{0}$ of finitely generated free algebras of $Θ$ is of great importance. In this paper, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety $_{R}\mathcal{M}$ of semimodules over an IBN-semiring $R$ (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of …

Uniformly Flat Semimodules

We revisit the notion of flatness for semimodules over semirings. In particular, we introduce and study a new notion of uniformly flat semimodules based on the exactness of the tensor functor. We also investigate the relations between this notion and other notions of flatness for semimodules in the literature.

关于半环的 Jacobson 根与半单性

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。

Finite simple additively idempotent semirings

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…

On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields

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.

三元$\Gamma$-半模中的Barr正合性与基于同余的同调代数

Barr-Exactness and Congruence-Based Homological Algebra in Ternary $Γ$-Semimodules

本文在三元$\Gamma$-半模范畴中建立同余优先的同调代数演算，定义同余意义下的核与商，证明态射典范分解与同余版同构定理，给出Barr-正合表述，并得到投射生成准则与经典图引理失效的结论。

Codensity, profiniteness and algebras of semiring-valued measures

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…

Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the e…