We construct a family of Σ-uniform Abelian groups and a family of Σ-uniform rings. Conditions are specified that are necessary and sufficient for a universal Σ-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 Σ-function exists in hereditarily finite admissible sets \( \mathbb{H}\mathbb{F}(G) \) and \( \mathbb{H}\mathbb{F}(K) \), where G = ⊕{Z p | p ∈ S} is a group, Z p is a cyclic group of order p, K = ⊕{F p | p ∈ S} is a ring, and F p is a prime field of characteristic p.
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Supported by RFBR (project No. 08-01-00336) and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-3606.2010.1).
Translated from Algebra i Logika, Vol. 51, No. 1, pp. 129-147, January-February, 2012.
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Khisamiev, A.N. Σ-Uniform structures and Σ-functions. II. Algebra Logic 51, 89–102 (2012). https://doi.org/10.1007/s10469-012-9172-y
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DOI: https://doi.org/10.1007/s10469-012-9172-y