Nonlinear Lie-type centralizers on incidence algebras

Let (Formula presented.) be a 2-torsion-free commutative ring with unity, X a locally finite preordered set, and (Formula presented.) the incidence algebra of X over (Formula presented.) with center (Formula presented.). In this paper, we provide a condition under which every nonlinear Lie n-centralizer (Formula presented.) on (Formula presented.) is proper. Under the same condition, we also show that every nonlinear generalized Lie n-derivation (Formula presented.) of (Formula presented.) can be written as (Formula presented.) where (Formula presented.) and D is a nonlinear Lie n-derivation.