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A structure is ultrahomogeneous if every isomorphism between finitely generated substructures can be extended to an automorphism of the structure. The aim of this paper is to provide an explicit description of ultrahomogeneous monounary algebras of arbitrary cardinalities. We also show that a monounary algebra is ultrahomogeneous if and only if it is 1-ultrahomogeneous. These results are then viewed in the context of a several related concepts, including ω -categoricity, homomorphism-homogeneity, (partial-)homogeneity, and transitivity.
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Thomas David Quinn-Gregson (2026) studied this question.
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