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arXiv · 2509.20726

Notes on the ordered set $A^A$. Part IV. The dual ${}^{A}\!A$ of $A^A$ for finite ordered sets

G. Grätzer
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Abstract

Let $A$ be a finite ordered set. Define the ordered set $A^A$ as the set of all maps from $A$ to $A$, ordered pointwise. Let ${}^{A} A$ be the dual of $A^A$. We prove results in the spirit of Parts~I--III, but now using both $A^A$ and ${}^{A}A$. For example, if \[ \Bigl({}^{{}^{ {}^{ {}^{A}A}A}A}A\Bigr)^{A^{A^{A}}} \] is isomorphic to \[ \Bigl({}^{ {}^{ {}^{ {}^{B}B}B}B}B\Bigr)^{B^{B^{B}}} \] for finite ordered sets $A$ and $B$, then $A$ is isomorphic to $B$.

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G. Grätzer. 2025-09-25. Notes on the ordered set $A^A$. Part IV. The dual ${}^{A}\!A$ of $A^A$ for finite ordered sets. https://arxiv.org/abs/2509.20726

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