  The queue number of a poset is the queue number of its cover graph when the
vertex order is a linear extension of the poset. Heath and Pemmaraju
conjectured that every poset of width $w$ has queue number at most $w$. The
conjecture has been confirmed for posets of width $w=2$ and for planar posets
with $0$ and $1$. In contrast, the conjecture has been refused by a family of
general (non-planar) posets of width $w>2$.
  In this paper, we study queue layouts of two-dimensional posets. First, we
construct a two-dimensional poset of width $w > 2$ with queue number $2(w -
1)$, thereby disproving the conjecture for two-dimensional posets. Second, we
show an upper bound of $w(w+1)/2$ on the queue number of such posets, thus
improving the previously best-known bound of $(w-1)^2+1$ for every $w > 3$.
