Reconstructing Finite Ordered Sets from Higher Exponentials

Abstract: For the finite ordered sets $A, D$, write $A^D$ for the ordered set of isotone maps $D \to A$ with the pointwise order. It was proved in earlier work that the order structure of $A^A$ determines~$A$ up to isomorphism. In this note we extend the result to higher function ordered sets such as $A^{(A^A)}$ and $(A^A)^A$. Our main theorem shows that the structure of $A^D$ determines~$A$.