Some mappings related to derivations on incidence algebras

Let (X,≤) be a locally finite pre-ordered set and R be a commutative ring with unity. We prove that if T is an R-linear mapping on the incidence algebra I(X,R), then the following are equivalent: (i) T is a generalized derivation; (ii) T is a local generalized derivation; (iii) fT(g)h=0 whenever f,g,h∈I(X,R) with fg=gh=0. We also show that every multiplicative derivation on I(X,R) with |X|≥2 is additive.