Hasse diagram of a total order not linear

Question

A total order is also called a linear order. But the Hasse diagram of a total order does not need to be a simple single line.

So the terms 'linear' and 'chain' are misleading.

Is this correct?

Answer 1

In a total order (also known as linear order), every two elements must be comparable, i.e. for all $a,b$, either $a \le b$ or $b \le a$. The link given in the comments above is to the Hasse diagram of the poset of subsets of a 3 element set, and while this relation is a partial order it is not a total order. For example, the elements $\{x,y\}$ and $\{x,z\}$ which lie in the same level set of the poset/Hasse diagram are not comparable because neither is a subset of the other.