If $L$ is a (fully defined) lattice, then given a subset $S \subseteq L$, it is known that the ideal $I(S)$ of $L$ generated by the set $S$ has the description $$ I(S) = \{x \in L : x \leq s_1 \vee \ldots \vee s_n, \mbox{with}\ s_i \in S\}.$$
Then if $P$ is a partially defined lattice and $S \subseteq P$, then it would seem natural that the ideal $I(S)$ of $P$ generated by $S$ would have the following description:
$$ I(S) = \{x \in P : \exists s_1, \ldots, s_n \in S \ \mbox{such that $s_1 \vee \ldots \vee s_n$ is defined and} \ x \leq s_1 \vee \ldots \vee s_n\}.$$
However, this does not seem to work, as the above set is not in general a partial lattice ideal, because it may not be closed under joins that exist. So does anyone know of the correct (bottom-up) description of the ideal generated by a set in a partial lattice?
In general an ideal I of an ordered set S is an upward directed lower set.
If A subset S, the ideal generated by A, I(A) is the smallest lower
set L containg A, with for all x,y in L, some b in L with x,y <= b.
Depending upon S, I(A) may or may not exist. In the case that S is
upward directed, I(A) will exist.