A difference on a poset $(P,\leq)$ is a partial binary
operation $\ominus$ on $P$ such that $b\ominus a$ is defined
if and only if $a\leq b$ subject to conditions $a\leq b\
\implies \ b\ominus (b\ominus a) = a$ and $a\leq b\leq c \
\implies \ (c\ominus a) \ominus(c\ominus b) = b\ominus a$. A
difference poset (DP) is a bounded poset with a difference.
A generalized difference poset (GDP) is a poset with a
difference having a smallest element and the property
$b\ominus a = c\ominus a \ \implies \ b = c$. We prove that
every GDP is an order ideal of a suitable DP, thus extending
previous similar results of Janowitz for generalized
orthomodular lattices and of Mayet-Ippolito for (weak)
generalized orthomodular posets. Various results and
examples concerning posets with a difference are included.
