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Varieties generated by completions

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Abstract

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the completions of representable relation algebras does not contain all relation algebras. This answers Problem 1.1(1) from Maddux’s 2018 Algebra Universalis paper in the negative. At the same time, we confirm the suggestion in that paper that the finite maximal relation algebras constructed in M. Frias and R. Maddux’s 1997 Algebra Universalis paper are not in the variety generated by the completions of representable relation algebras. We prove that there are continuum many varieties between the variety generated by the completions of representable relation algebras and the variety of relation algebras.

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  1. Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences, Reáltanoda st. 13-15, Budapest, 1053, Hungary

    Hajnal Andréka & István Németi

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  1. Hajnal Andréka
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  2. István Németi
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Correspondence to Hajnal Andréka.

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Andréka, H., Németi, I. Varieties generated by completions. Algebra Univers. 80, 30 (2019). https://doi.org/10.1007/s00012-019-0602-8

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