Ultrafilter extensions of linear orders
Abstract
It was recently shown that arbitrary first-order models canonically extend to models (of the same language) consisting of ultrafilters. The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a technique allowing 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 not more an order however 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.
Cite
@article{arxiv.1310.4533,
title = {Ultrafilter extensions of linear orders},
author = {Denis I. Saveliev},
journal= {arXiv preprint arXiv:1310.4533},
year = {2013}
}