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This is a continuation of our earlier works \cite{KhrypchenkoWei, Yang20211,Yang20212} with respect to (non-)linear Lie-type derivations of finitaryincidence algebras. Let $X$ be a pre-ordered set, $\mathcal{R}$ be a$2$-torsionfree and $(n-1)$-torsionfree commutative ring with identity, where$n\geq 2$ is an integer. Let $FI(X,\mathcal{R})$ be the finitary incidencealgebra of $X$ over $\mathcal{R}$. In this paper, a complete clarification isobtained for the structure of nonlinear Lie-type derivations of$FI(X,\mathcal{R})$. We introduce a new class of derivations on$FI(X,\mathcal{R})$ named inner-like derivations, and prove that each nonlinearLie $n$-derivation on $FI(X,\mathcal{R})$ is the sum of an inner-likederivation, a transitive induced derivation and a quasi-additive induced Lie$n$-derivation. Furthermore, if $X$ is finite, we show that a quasi-additiveinduced Lie $n$-derivation can be expressed as the sum of an additive inducedLie derivation and a central-valued map annihilating all $(n-1)$-thcommutators. We also provide a sufficient and necessary condition such thatevery nonlinear Lie $n$-derivation of $FI(X,\mathcal{R})$ is of proper form.Some related topics for further research are proposed in the last section ofthis article.

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics