AI 中文总结
该论文刻画关联代数上满足双侧零积处二次泛函恒等式的线性映射形式，证明其为标准形式的充要条件，涉及可比图2-连通性刻画与关联代数的双侧零积确定性质。
AI 中文摘要
设R为含单位元的交换环且满足1/2∈R，X为连通有限偏序集且|X|>2，I(X,R)为X上取值于R的关联代数。本文刻画满足F₁(f)g + fF₂(g) + F₃(g)f + gF₄(f)=0（当fg=gf=0时）的线性映射F₁,F₂,F₃,F₄:I(X,R)→I(X,R)的形式。我们证明：F_i均为所谓标准形式当且仅当X的可比图中任意两条边都包含在一个环内。证明的要素包括可比图中2-连通性的刻画以及关联代数的双侧零积确定性质。
英文摘要
Let $R$ be a commutative ring with unity such that $\frac{1}{2}\in R$. Let $X$ be a connected finite poset with $|X|>2$ and $I(X,R)$ be the incidence algebra of $X$ over $R$. In this paper, we characterize the forms of linear maps $F_1,F_2,F_3,F_4:I(X,R)\to I(X,R)$ satisfying \[ F_1(f)g+fF_2(g)+F_3(g)f+gF_4(f)=0, \] whenever $fg=gf=0$. We prove that the $F_i$'s are of the so-called standard form if and only if any two edges in the comparability graph of $X$ are contained in one cycle. The ingredients of the proof contain a characterization of $2$-connectedness in comparability graph and the two-sided zero product determined property of incidence algebras.
Comments23 pages
