A solution to a spectral poset problem of Lewis and Ohm

We show that there is a 1-dimensional (countable) non-spectral poset X such that for all x ≠ y ∈ X , ↑ x ∩ ↑ y and ↓ x ∩ ↓ y are finite subsets. On the other hand, we obtain some sufficient conditions for posets to be spectral.