Is it true that Hasse diagram of ToSET(totally ordered set) is vertical line?

Question

I am learning hasse diagram concept. I read that toset means poset + every pair of elements in set must be comparable(ie.., related). If any pair of elements are comparable then there will not be branching right(pls correct me if wrong). If there is branching (say V shape) then 2 top elements on top of V-shape will not be comparable. THen i think is it true that toset will always be a vertical line? pls correct me if wrong.

Answer 1

You are correct that a finite toset can be represented by a entirely vertical Hasse diagram as you describe. For infinite sets it becomes debatable there even is a Hasse diagram, although the vertical intuition is still a good one since it works "locally" (i.e. for finite subtosets). Indeed, this is the reason another term for totally ordered is "linearly" ordered - because our intuition is that it can exist on a line. The Hasse diagram of a toset of size $n$ is the corresponding "line graph."

For infinite tosets, we can have vertices that are not connected by edges and edges necessarily of arbitrarily small length - for instance the ordinal $\omega+1$, which you can think of as $\mathbb{N}\sqcup\{\infty\}$, or more topologically as $\{-1/n\mid n>0\}\sqcup\{0\}$. This also shows the cardinality does not determine the diagram, since $\omega$ itself ought to have a different diagram. It may not even make sense to say there are edges, for instance what would the Hasse diagram of $\mathbb{Q}$ be? And the toset may not even embed in $\mathbb{R}$ at all, for instance if its cardinality is greater than $\mathfrak{c}=|\mathbb{R}|$.