Algebraic structures from the point of view of complete multiplicative lattices

Multiplicative structure of 2x2 tropical matrices

We study the algebraic structure of the semigroup of all $2 \times 2$ tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups of this semigroup.

乘性李代数非交换扩张的对应序列、Wells序列与Hochschild-Serre序列

Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras

该研究将群论中的三个经典结果推广到乘性李代数的非交换二阶上同调场景，证明了对应定理、Wells定理并建立一维Lyndon-Hochschild-Serre正合序列，特殊情形下可复现已有结论。

Approximately Groups in Proximal Relator Spaces: An Algebraic View of Digital Images

The focus of this article is to define the descriptively approximations in proximal relator spaces. Afterwards, descriptive approximately algebraic structures such as groupoids, semigroups and groups in digital images endowed with descriptive Efremovic proximity relations were introduced.

Degree One Milnor $K$-invariants of Groups of Multiplicative Type

Let $G$ be a commutative affine algebraic group over a field $F$, and let $H \colon \mathrm{Fields}_{F} \to \mathrm{AbGrps}$ be a functor. A (homomorphic) $H$-invariant of $G$ is a natural transformation $\mathrm{Tors}(-, G) \to H$, where $\mathrm{Tors}(-, G)$ is the functor $\mathrm{Fields}_{F} \to \mathrm{AbGrps}$ taking a field extension $L/F$ to the group of isomorphism classes of $G_{L}$-torsors over $\mathrm{Spec}(L)$. The goal of this paper is to compute the group $\mathrm{Inv}_{\mathrm{hom}}^{1}(G, H)$ of $H$-invariants of $G$ when $G$ is a group of multiplicative type, and $H$ is the …

极大环面与代数群的Cartan子群

Maximal toroids and Cartan subgroups of algebraic groups

本文提出了一种适用于所有一次代数群的统一理论，探讨了极大环面与Cartan子群之间的关系，并推广了已知结果至更广泛的情况。

On Tensor Products of Simple Modules for Simple Groups

In an attempt to get some information on the multiplicative structure of the Green ring we study algebraic modules for simple groups, and associated groups such as quasisimple and almost-simple groups. We prove that, for almost all groups of Lie type in defining characteristic, the natural module is non-algebraic. For alternating and symmetric groups, we prove that the simple modules in $p$-blocks with defect groups of order $p^2$ are algebraic, for $p\leq 5$. Finally, we analyze nine sporadic groups, finding that all simple modules are algebraic for various primes and sporadic groups