Generalized Lie Derivations of Incidence Algebras

Let [Formula: see text] be a 2-torsion-free commutative ring with unity, [Formula: see text] a locally finite pre-ordered set and [Formula: see text] the incidence algebra of [Formula: see text] over [Formula: see text]. We prove that under certain conditions every generalized Lie derivation and every [Formula: see text]-Lie derivation (with a linear map [Formula: see text]) of [Formula: see text] is the sum of a left multiplication map, a derivation and a central map which vanishes on all commutators of [Formula: see text].