This article is cited in
2 papers
Universal functions and $\Sigma_{\omega}$-bounded structures
A. N. Khisamiev Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Abstract:
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.
Keywords:
admissible set, $\Sigma$-function, universal $\Sigma$-function hereditarily finite superstructure, tree.
UDC:
512.540+
510.5 Received: 08.04.2020
Revised: 24.08.2021
DOI:
10.33048/alglog.2021.60.207