Year: 2026
Type: preprint
Abstract: For ordered sets $X$ and $Y$, let $Y^X$ denote the ordered set of order-preserving maps from $X$ to $Y$, ordered pointwise. In 1984, D. Duffus asked whether $A^A\cong B^B$ forces $A\cong B$ when $A$ and $B$ are finite ordered sets. We prove that it does. The proof combines direct decompositions with an auxiliary ordering of connected ordered sets and a careful choice of intervals of maximum height. We then extend the reconstruction result to several forms of repeated exponentiation.
Source: arXiv (Cornell University)
Authors: Farley, Jonathan David, Grätzer, George
Institutions: University of Manitoba, Morgan State University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222443687