  In 2010, Bousquet-M\'elou et al. defined sequences of nonnegative integers
called ascent sequences and showed that the ascent sequences of length $n$ are
in one-to-one correspondence with the interval orders, i.e., the posets not
containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating
functions, this provided an answer to the longstanding open question of
enumerating the (unlabeled) interval orders. A semiorder is an interval order
having a representation in which all intervals have the same length. In terms
of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and
$\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has
long been known to be the $n$-th Catalan number. However, describing the ascent
sequences that correspond to the semiorders under the bijection of
Bousquet-M\'elou et al. has proved difficult. In this paper, we discuss a major
part of the difficulty in this area: the ascent sequence corresponding to a
semiorder may have an initial subsequence that corresponds to an interval order
that is not a semiorder.
  We define the hereditary semiorders to be those corresponding to an ascent
sequence for which every initial subsequence also corresponds to a semiorder.
We provide a structural result that characterizes the hereditary semiorders and
use this characterization to determine the ordinary generating function for
hereditary semiorders. We also use our characterization of hereditary
semiorders and the characterization of semiorders of dimension 3 given by
Rabinovitch to provide a structural description of the semiorders of dimension
at most 2. From this description, we are able to determine the ordinary
generating for the semiorders of dimension at most 2.
