A note on dual Dedekind finiteness

A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. It is proved consistent with $\mathsf{Z}\mathsf{F}$ (i.e., the Zermelo–Fraenkel set theory without the axiom of choice) that there exists a family $⟨A_n{⟩}_{n\in \omega }$ of sets such that, for all $n\in \omega$, $A_n^n$ is dually Dedekind finite whereas $A_n^{n+1}$ is dually Dedekind infinite. This resolves a question that was left open in [J. Truss, Fund. Math. 84, 187–208 (1974)].