Geodesic
Universal functions and $\Sigma_{\omega}$-bounded structures
Algebra i logika, Tome 60 (2021) no. 2, pp. 210-230.

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We introduce the notion of a $\Sigma_{\omega}$-bounded structure and specify a necessary and sufficient condition for a universal $\Sigma$-function to exist in a hereditarily finite superstructure over such a structure, for the class of all unary partial $\Sigma$-functions assuming values in the set $\omega$ of natural ordinals. Trees and equivalences are exemplified in hereditarily finite superstructures over which there exists no universal $\Sigma$-function for the class of all unary partial $\Sigma$-functions, but there exists a universal $\Sigma$-function for the class of all unary partial $\Sigma$-functions assuming values in the set $\omega$ of natural ordinals. We construct a tree $T$ of height $5$ such that the hereditarily finite superstructure ${\mathbb {HF}}(T)$ over $T$ has no universal $\Sigma$-function for the class of all unary partial $\Sigma$-functions assuming values $0, 1$ only.
Mots-clés : admissible set
Keywords: $\Sigma$-function, universal $\Sigma$-function hereditarily finite superstructure, tree. 
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A. N. Khisamiev. Universal functions and $\Sigma_{\omega}$-bounded structures. Algebra i logika, Tome 60 (2021) no. 2, pp. 210-230. http://geodesic.mathdoc.fr/item/AL_2021_60_2_a6/

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