Supercommuting Maps on Incidence Algebras with Superalgebra Structure

Let R bea2-torsion free and n!-torsion free commutative ring with unity, and let X bealocally finite pre-ordered set. Using the method of Ghahramani and Heidari Zadeh [22], we endow the incidence algebra I(X,R) with a superalgebra structure via a nontrivial idempotent, decomposing I(X,R) into even and odd parts A0⊕A1. If any two directed edges in each connected component of the complete Hasse diagram (X,D) are contained in one cycle, then every supercommuting map on the incidence algebra I(X,R) is proper. A supercommuting map θ : I(X,R) → I(X,R) satisfies [θ(x),x]s = 0 for all x ∈ I(X,R), where [a,b]s = ab − (−1)|a||b|ba is the supercommutator for homogeneous elements, extended linearly [40, 41, 27]. We prove that such maps are of 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, under the given cycle condition, generalizing results on commuting maps [24, 44, 14].