Countable homogeneous linearly ordered posets

A relational structure is called homogeneous if each isomorphism between its finite substructures extends to an automorphism of that structure. A linearly ordered poset is a relational structure consisting of a partial order relation on a set, along with a total (linear) order that extends the partial order in question. We characterise all countable homogeneous linearly ordered posets, thus extending earlier work by Cameron on countable homogeneous permutations. As a consequence of our main result it turns out that, up to isomorphism, there is a unique homogeneous linear extension of the random poset, the unique countable homogeneous universal partially ordered set.