I have 2 questions
1) Should totally ordered set always contain only 2 elements ?
I think that a poset with a single element is also totally ordered as it can be reflexive, anti-symmetric and transitive. Also it is comparable with itself. My professor says that for totally ordered set you always need 2 elements (a and b) for comparison.
2) Are totally ordered set always complemented lattice ?
I think that there might be a totally ordered set which is infinite and infinite sets have no upper bound. So it's not complemented lattice
Am I correct?
Indeed, a poset with just one element is not only totally ordered, it even is well ordered. Also note that the empty set with its only possible relation also is totally ordered (and well ordered).
If that were not the case, the theory of ordinals would be in deep trouble. :-)
It is true that there are totally ordered sets that are infinite, but it is not true that infinite sets never have upper bounds. For example the real closed interval $[0,1]$ with the usual order is both totally ordered and infinite, but clearly has an upper bound (namely $1$).
However it is true that there exist infinite totally ordered sets that do not have an upper bound; $\mathbb N$ is the simplest example of it.