Amorphous sets and dual Dedekind finiteness

Abstract

A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists an amorphous set $A$ whose power set $\mathscr{P}(A)$ is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss [J. Truss, Fund. Math. 84, 187--208 (1974)]. Nevertheless, we prove in $\mathsf{ZF}$ that, for all strictly amorphous sets $A$ and all natural numbers $n$, $\mathscr{P}(A)^n$ is dually Dedekind finite, which generalizes a result of Goldstern.