Countable homogeneous linearly ordered posets (Q449222)

A linearly ordered poset is a structure \((A,\prec,\sqsubset)\), where \((A,\prec)\) is a poset and \((A,\sqsubset)\) is a linear extension of \((A,\prec)\).NEWLINENEWLINE The authors characterize all countable homogeneous linearly ordered posets. The discussion splits into two basic cases. The first case leads to permutations considered as structures with two linear orderings. The countable homogeneous permutations were classified by Cameron. In the non-permutational case the authors obtain two additional families: The first can be thought of as a mixture of \(k\) (where \(2 \leq k \leq \aleph_0\)) copies of \((\mathbb{Q},<)\) shuffled into a singular linear order, while the second has a single member, the random lo-poset.