Let $(X, \leq)$ be a bidirected poset (for every pair of elements in $X$, they share a common succesor and a common predecessor). If $X$ is a total order, then trivially there is a chain connecting any two pairs of elements in $X$. Is the inverse true?
Suppose that in a bidirected poset $(X, \leq)$, there is a chain connecting any two pairs of elements. Is $(X, \leq)$ a total order?
I think this is true but I'm failing to find a proof or a counterexample.
Partial orders are transitive. So if for any pair of elements there is a chain connecting the two, then any two elements are comparable. If you have a poset where any two elements are comparable, you have a total order.