  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.
