Invertibility preservers of finitary incidence algebras

Let X be an arbitrary poset and K an arbitrary field. We describe linear unital invertibility preservers of the finitary incidence algebra FI(X, K) in terms of certain maps of the power set algebra \({\mathcal {P}}(X)\) $\mathcal{P}(X)$ and linear maps \(FI(X,K)\rightarrow J(FI(X,K))\) $FI(X,K)\to J(FI(X,K))$ .