Notes on the ordered set $A^A$. Part I. The classical problem

TL;DR

This paper proves that if the ordered sets of isotone self-maps of two finite ordered sets are isomorphic, then the original sets themselves are isomorphic, resolving a longstanding question in order theory.

Contribution

It establishes that isomorphism of the ordered sets of isotone self-maps implies the isomorphism of the original finite ordered sets, answering a question from 1978.

Findings

Isomorphic ordered sets of self-maps imply original sets are isomorphic.

Resolves a question posed by D. Duffus in 1978.

Related results by Duffus--Wille and Farley are discussed.

Abstract

Let $A$ and $B$ be finite ordered sets. We show that if the ordered sets of isotone self-maps $A^A$ and $B^B$ (ordered pointwise) are isomorphic, then $A$ and $B$ are isomorphic. This resolves a question originating with D. Duffus in 1978, with related results by D. Duffus--R. Wille in 1979 and J. Farley in 2023.