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The persistence of universal formulae in free algebras. (English) Zbl 0607.08005

The authors prove the equivalence of the following two properties of free algebras of a nontrivial variety V: (i) V is discriminated by its free algebra of rank r; (ii) for any \(s\geq r\) free algebras of rank r and s satisfy the same universal sentences.
They also introduce a generalization of discrimination and find a logical concept equivalent to it. Finally, they indicate some particular cases in varieties of groups where discrimination and strong discrimination coincide.
Reviewer: Yu.A.Bakhturin

MSC:

08B20 Free algebras
20E10 Quasivarieties and varieties of groups
03C05 Equational classes, universal algebra in model theory
20A15 Applications of logic to group theory
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References:

[1] DOI: 10.1007/BF01226028 · Zbl 0406.20024 · doi:10.1007/BF01226028
[2] DOI: 10.1007/BF01111331 · Zbl 0125.01402 · doi:10.1007/BF01111331
[3] Bell, Models and Ultraproducts (1971)
[4] Grätzer, Universal Algebra (1968)
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