Higher Derivations of 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…

Jordan Derivations of Incidence Algebras

Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan derivation of $I(X,\mathcal{R})$ is a derivation provided that $\mathcal{R}$ is $2$-torsion free.

关联环：自同构与导子

Incidence Rings: Automorphisms and Derivations

本文研究预序集$X$与交换环$T$上代数$R$构成的关联代数$I(X,R)$的自同构、同构及导子性质。

Lie Derivations of Incidence Algebras

Let $X$ be a locally finite preordered set, $\mathcal R$ a commutative ring with identity and $I(X,\mathcal R)$ the incidence algebra of $X$ over $\mathcal R$. In this note we prove that each Lie derivation of $I(X,\mathcal R)$ is proper, provided that $\mathcal R$ is $2$-torsion free.

关联代数的Jordan *-导子

Jordan $*$-derivations of incidence algebras

本文研究局部有限偏序集上关联代数的 Jordan *-导子，证明其均为内 *-导子且为转置 Jordan *-导子，并给出存在非 *-导子的 Jordan *-导子的结论。

Isomorphisms and derivations of partial flag incidence algebras

Let $P$ and $Q$ be finite posets and $R$ a commutative unital ring. In the case where $R$ is indecomposable, we prove that the $R$-linear isomorphisms between partial flag incidence algebras $I^3(P,R)$ and $I^3(Q,R)$ are exactly those induced by poset isomorphisms between $P$ and $Q$. We also show that the $R$-linear derivations of $I^3(P,R)$ are trivial.

关联余代数：自同构与导子

Incidence Coalgebras: Automorphisms and Derivations

本文利用关联余代数 Co(X,F) 的对偶代数同构于关联代数 I(X,F) 这一性质，刻画了域上偏序集关联余代数的自同构与导子。

关联代数的斜导子

Skew derivations of incidence algebras

本文研究局部有限偏序集上关联代数的斜导子，证明其可分解，且斜导子空间模内斜导子子空间与第一上同调群同构。

Lie Triple Derivations of Incidence Algebras

Let $\mathcal{R}$ be a $2$-torsion free commutative ring with unity, $X$ a locally finite pre-ordered set and $I(X,\mathcal{R})$ the incidence algebra of $X$ over $\mathcal{R}$. If $X$ consists of a finite number of connected components, we prove in this paper that every Lie triple derivation of $I(X,\mathcal{R})$ is proper.

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。

代数Toeplitz代数的李理想与李导子

Lie Ideals and Lie Derivations of the Algebraic Toeplitz Algebra

本文对代数Toeplitz代数$\boldsymbol{\top}$的李理想进行分类，证明非中心李理想包含有限迹零李代数$\boldsymbol{s}$，还将其李导子表示为结合导子与中心李导子之和。