I'm learning lattice theory by myself and I'm puzzled by the following paragraph (from Oligopoly pricing, X. Vives, p.18)
"A lattice ($S,\geq $) is complete if every nonempty subset of S has a supremum and an infimum in S. Any compact interval of the real line with the usual order is a complete lattice while the open interval (a,b) is a lattice but is not complete (indeed the supremum of (a,b) does not belong to (a,b))." But then the extremum does not belong to the interval, does it?
My intuition was $(-\infty,\infty)$ and $[5,7]$ are a lattice but $(5,7)$ is not.
What is wrong, what is right above, please?
Lattices don't in general need to have maximal or minimal elements, so e.g. $(5, 7)$ is indeed a lattice. It's not complete - any complete lattice does indeed have a top and a bottom - but that's fine. Remember that a lattice just has to be closed under meets/joins of finite sets.