代数结构III上的Algebraic geometry：方程Noether性质和compactness

Ershov代数的方程Noether性质

Equationally Noetherian property of Ershov algebras

本文研究Ershov代数的方程Noether性和弱方程Noether性。本文给出了Ershov代数上方程组的两个标准形以及方程Noether性质和弱Noether性质的两个判据。

Representations via differential algebras and equationally Noetherian algebras

We show that free algebras of the variety of algebras generated by the Witt algebra $W_n$, the left-symmetric Witt algebra $L_n$, and the symplectic Poisson algebra $P_n$ can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that $W_n$, $L_n$, $P_n$, and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian.

On equations over direct powers of algebraic structures

We study systems of equations over graphs, posets and matroids. We give the criteria, when a direct power of such algebraic structures is equationally Noetherian. Moreover we prove that any direct power of a finite algebraic structure is weakly equationally Noetherian.

Equations over direct powers of algebraic structures in relational languages

We study equations over relational structures that approximate groups and semigroups. For such structures we proved the criteria, when a direct power of such algebraic structures is equationally Noetherian.

相干条件：任意代数类的代数几何

Coherent Conditions: Algebraic Geometry for Arbitrary Classes of Algebras

该研究将通用代数几何推广到任意代数类的谱理论，建立无量词命题与Zariski闭集的对应，并探讨方程Noether类与不可约谱。

交换对称幺半范畴上的Noether概型

Noetherian Schemes over abelian symmetric monoidal categories

本文发展了交换对称幺半范畴范畴上代数几何的基本结果。设$A$是满足一定条件的交换对称幺半范畴$(\mathbf C,\otimes,1)$中的交换幺半群对象，令$\mathcal E(A)=Hom_{A-Mod}(A,A)$ .如果$A$的子代数满足某种紧性，我们称$A$是Noether的.我们研究了$A$相对于任何$s\in \mathcal E(A)$的局部化，并定义了$A$相对于任何理想$\mathscr I\subseteq \mathcal E(A)$的商$A/\mathscr I$。我们用它来开发适当的类似物的基本概念，从通常的代数几何（如诺特计划，不可约，积分和约化概形，功能领域，在一般的一个闭子概形点的局部环等）的计划在$(\mathbf C,\otimes,1)$。我们关于对称幺半范畴$(\mathbf C,\otimes,1)$上概型的概念是Toën和Vaquié的概念。