An example Ginsburg said in 1984 he was "unable to find" and a forbidden subposet characterization of subsets of regular posets

In 1984, Ginsburg wrote, "We have been unable to find an example of an ordered set $P$ having the properties of [being complete, densely ordered, with no antichain other than $\{0\}$ and $\{1\}$ that is a cutset] and in which all antichains are countable." In this very brief note, such an example is shown. Posets that can be embedded in regular posets are characterized as posets that do not contain $\omega\times\{0,1\}$ or its dual as a subposet. Any such poset $P$ can be embedded in a regular poset that can be embedded in any other regular poset containing $P$.