Supercommuting Maps on Incidence Algebras with Superalgebra Structures

Let R be a 2-torsion-free and n!-torsion-free commutative ring with unity, and let X be a locally finite preordered set. We endow the incidence algebra I(X,R) with a superalgebra structure via a nontrivial idempotent, which decomposes I(X,R) into even and odd parts A0⊕A1. Our main result shows that if any two directed edges in each connected component of the complete Hasse diagram (X,D) lie in one cycle, then every supercommuting map on I(X,R) is proper. A supercommuting map θ:I(X,R)→I(X,R) is defined by the condition [θ(x),x]s=0 for all x∈I(X,R), where [a,b]s=ab−(−1)|a||b|ba is the supercommutator. We prove that such maps must take the form θ(x)=λx+μ(x), where λ∈Zs(I(X,R)) (the supercenter) and μ:I(X,R)→Zs(I(X,R)) is an R-linear map. This generalizes the known results on commuting maps of incidence algebras and other associative algebras.