Notes on the ordered set $A^A$. Part III. Exponent-cancellations

We continue the study of exponent-cancellation for finite ordered sets. It is known that $A$ can be reconstructed from $A^{A}$, from $(A^{A})^{A}$, and from $A^{A^{A}}$. In this note we prove the next result in this hierarchy: the ordered set $(A^{A^{A}})^{A^{A^{A}}}$ determines $A$ up to isomorphism.