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arXiv · 2607.02725

A note on ultrahomogeneous unary algebras

Cheng Liao
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Abstract

In a recent paper \cite{q}, Quinn-Gregson fully classified ultrahomogeneous mono-unary algebras. In particular, he proved that every locally finite mono-unary algebra with finitely many 1-orbits is ω-categorical, and every 1-ultrahomogeneous mono-unary algebra is ultrahomogeneous. He then asked if these two results can be generalized to unary algebras. This short note answers the second question negatively by providing a simple counterexample. We also show that the first question has a positive answer in ``tree-like" cases which covers his result to mono-unary algebras and has a small combinatorial implication.

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Cheng Liao. 2026-07-09. A note on ultrahomogeneous unary algebras. https://arxiv.org/abs/2607.02725

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