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Ultrafilter Extensions of Linearly Ordered Sets

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Abstract

Ultrafilter extensions of arbitrary first-order models were introduced in Saveliev (2012). The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a technique that was used to obtain significant results in algebra and dynamics. Here we consider another particular case where the models are linearly ordered sets. We explicitly calculate the extensions of a given linear order and the corresponding operations of minimum and maximum on a set. We show that the extended relation is no longer an order though it is close to the natural linear ordering of nonempty half-cuts of the set and that the two extended operations define a skew lattice structure on the set of ultrafilters.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkina, Moscow, 119991, Russia

    Denis I. Saveliev

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  1. Denis I. Saveliev
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Correspondence to Denis I. Saveliev.

Additional information

Partially supported by RFBR grants 11-01-00958 and 11-01-93107 and ARRS grant P1-0288. Avalaible at arXiv:1310.4533 [math.LO].

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Saveliev, D. Ultrafilter Extensions of Linearly Ordered Sets. Order 32, 29–41 (2015). https://doi.org/10.1007/s11083-013-9313-5

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  • DOI: https://doi.org/10.1007/s11083-013-9313-5

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