Poisson Structures on Finitary Incidence Algebras

Jordan derivations of finitary incidence rings

Let $P$ be a preordered set, $R$ a ring and $FI(P,R)$ the finitary incidence ring of $P$ over $R$. We find a criterion for all Jordan derivations of $FI(P,R)$ to be derivations and generalize Theorem 3.3 from arXiv:1411.6123. In particular, we prove that each Jordan derivation of the ring $RFM_I(R)$ of row-finite $I\times I$-matrices over $R$ is a derivation, if $|I|>1$.

有限关联代数的可逆性保持映射

Invertibility preservers of finitary incidence algebras

该研究针对任意偏序集$X$与任意域$K$，通过幂集代数映射与线性映射刻画了有限关联代数$FI(X,K)$的线性单位可逆性保持映射，丰富了关联代数的映射理论。

Local automorphisms of finitary incidence algebras

Let $R$ be a commutative, indecomposable ring with identity and $(P,\le)$ a partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le)$ over $R$. We will show that, in most cases, local automorphisms of $FI(P)$ are actually $R$-algebra automorphisms. In fact, the existence of local automorphisms which fail to be $R$-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result.

Anti-isomorphisms and involutions on the idealization of the incidence space over the finitary incidence algebra

Let $K$ be a field and $X$ a partially ordered set (poset). Let $FI(X,K)$ and $I(X,K)$ be the finitary incidence algebra and the incidence space of $X$ over $K$, respectively, and let $D(X,K)=FI(X,K)(+)I(X,K)$ be the idealization of the $FI(X,K)$-bimodule $I(X,K)$. In the first part of this paper, we show that $D(X,K)$ has an anti-automorphism (involution) if and only if $X$ has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on $D(X,K)$. In the second part, we obtain the classification of involutions on $D(X,K)$ to the case when …

有限入射代数上的第二类对合

Involutions of the second kind on finitary incidence algebras

本文研究有限入射代数上的第二类对合，给出了其存在的充要条件、特征描述，以及在特定条件下两个此类对合等价的判定条件。

Finitary incidence algebras

We consider the functions in two variables on an arbitrary poset, for which the convolution operation is defined. We obtain the generalization of incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.

有限关联代数及其单位群的恒等式与导出长度

Identities and derived lengths of finitary incidence algebras and their group of units

本文研究了域$K$上偏序集$X$的有限关联代数$FI(X,K)$，确定了其满足多项式恒等式及单位群满足群恒等式的条件，并求出了其李导出长度与单位群的导出长度。

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

This is a continuation of our earlier works \cite{KhrypchenkoWei, Yang20211, Yang20212} with respect to (non-)linear Lie-type derivations of finitary incidence algebras. Let $X$ be a pre-ordered set, $\mathcal{R}$ be a $2$-torsionfree and $(n-1)$-torsionfree commutative ring with identity, where $n\geq 2$ is an integer. Let $FI(X,\mathcal{R})$ be the finitary incidence algebra of $X$ over $\mathcal{R}$. In this paper, a complete clarification is obtained for the structure of nonlinear Lie-type derivations of $FI(X,\mathcal{R})$. We introduce a new class of derivations on $FI(X,\mathcal{R})$ na…

李关联代数上的转置Poisson结构

Transposed Poisson structures on Lie incidence algebras

本文研究了特征零域上有限连通偏序集关联代数李代数结构，证明了$\ rac{1}{2}$-导数的分解定理，并据此得出任意转置Poisson结构可分解为Poisson型、变异型和特定映射确定的结构之和。