The model companion of width-two orders (Q1267599)

Let \(W_n\) denote the universal class of all width-\(n\) orders. Building on work of Pouzet the authors prove that the class of existentially closed members of \(W_2\), \(W^{\text{ec}}_2\), is a first-order class. In fact for \(A\in W_2\), \(A\in W^{\text{ec}}_2\) if and only if \(A\) is a linear sum of d-homogeneous width-2 orders. As corollaries of their characterization they have that the model companion of \(W_2\) exists and it is complete, decidable, non-finitely axiomatizable, and has \(2^{\aleph}_0\) countable models. Another corollary is that \(W_2\) has a decidable universal theory.