Higher derivations of finitary incidence algebras (Q2288135)

A sequence of additive maps \((d_n)_{n\in \mathbb N}\) on a unital ring \(R\) is called a higher derivation if the identities \[ d_0(x)=x\quad \hbox{ and } \quad d_n(xy) = \sum_{k=0}^n d_k(x)d_{n-k}(y)\] hold. Examples include the sequence of additive maps \(d_n\colon x\mapsto r^{n-1}(rx-xr)\), with an element \(r\in R\) kept fixed, as well as, when \(R\) is an algebra over a field with characteristic \(0\), the sequence \((\frac{1}{n!}d^n)_{n\in\mathbb N}\) with \(d\colon R\to R\) being a usual derivation on \(R\). In fact, higher derivations are in one-to-one, onto correspondence with those automorphisms \(\alpha\) of the ring of formal power series \(R[[t]]\) which fix an indeterminate \(t\) and map each \(x\in R\subseteq R[[t]]\) into the set \(x+t R[[t]]\); the correspondence is given by \(\alpha(x)=\sum d_{n}(x)t^n\); \(x\in R\subseteq R[[t]]\). The main result of the paper under review describes the form of \(R\)-linear higher derivations on finitary incidence algebras \(FI(R)\) over commutative unital rings \(R\). Here, by definition, \(FI(R)\) is an \(R\)-algebra of \(R\)-valued functions with domain consisting of all pairs \((x,y)\), ordered in a given preordered set \(P\), which have a finite support when restricted to each of the subsets \(\Omega_{(x,y)}:=\{(u,v)\in P^2;\;\; x\le u<v\le y\}\). The \(R\)-module structure on \(FI(R)\) is standard and the multiplication is convolution-like \[(f\ast g)(x,y):=\sum_{x\le z\le y} f(x,z)g(z,y).\]