Extensions of ordered sets having the finite cutset property (Q1076700)

The embeddability of an ordered set P in an ordered set having the finite cutset property is studied in the paper. One of the main results is a necessary condition of embeddability stating that an uncountable subset of P has distinct elements x, p and q such that \(x^+\cap p^+=x^+\cap q^+\), where \(x^+=\{p\in P|\) \(x\leq p\}\). Another is an example of an ordered set which has no extension having the finite cutset property but every countable subset of which has such an extension.