Geodesic
$\Sigma$-uniform structures and $\Sigma$-functions. II
Algebra i logika, Tome 51 (2012) no. 1, pp. 129-147

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We construct a family of $\Sigma$-uniform Abelian groups and a family of $\Sigma$-uniform rings. Conditions are specified that are necessary and sufficient for a universal $\Sigma$-function to exist in a hereditarily finite admissible set over structures in these families. It is proved that there is a set $S$ of primes such that no universal $\Sigma$-function exists in hereditarily finite admissible sets $\mathbb{HF}(G)$ and $\mathbb{HF}(K)$, where $G=\oplus\{Z_p\mid p\in S\}$ is a group, $Z_p$ is a cyclic group of order $p$, $K=\oplus\{F_p\mid p\in S\}$ is a ring, and $F_p$ is a prime field of characteristic $p$.
Keywords: hereditarily finite admissible set, $\Sigma$-definability, universal $\Sigma$-function, $\Sigma$-uniform structure, Abelian group, ring. 
@article{AL_2012_51_1_a5,
     author = {A. N. Khisamiev},
     title = {$\Sigma$-uniform structures and $\Sigma${-functions.~II}},
     journal = {Algebra i logika},
     pages = {129--147},
     publisher = {mathdoc},
     volume = {51},
     number = {1},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2012_51_1_a5/}
}
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A. N. Khisamiev. $\Sigma$-uniform structures and $\Sigma$-functions. II. Algebra i logika, Tome 51 (2012) no. 1, pp. 129-147. http://geodesic.mathdoc.fr/item/AL_2012_51_1_a5/