 Solution

The two-argument version of Fn forcing consists of finite partial functions:Thus a stronger condition specifies more values. The empty partial function is the greatest condition. Two conditions are compatible forcing conditions exactly when they agree on their common domain; if they do, their set union is a common strengthening. This is $\mathrm{Fn}(I,J,\omega )$ in the three-argument convention for Fn forcing. If $I$ or $J$ is empty, only the empty condition exists. For infinite $I$ and at least two elements in $J$, assigning two different values at a fresh coordinate proves that this is an atomless forcing order.