  Given a poset $P$, a join-specification $\mathcal U$ for $P$ is a set of
subsets of $P$ whose joins are all defined. The set $\mathcal I_{\mathcal U}$
of downsets closed under joins of sets in $\mathcal U$ forms a complete
lattice, and is, in a sense, the free $\mathcal U$-join preserving
join-completion of $P$. The main aim of this paper is to address two questions.
First, given a join-specification $\mathcal U$, when is $\mathcal I_{\mathcal
U}$ a frame? And second, given a poset $P$, what is the structure of its set of
frame-generating join-specifications?
  To answer the first question we provide a number of equivalent conditions,
and we use these to investigate the second. In particular, we show that the set
of frame-generating join-specifications for $P$ forms a complete lattice
ordered by inclusion, and we describe its meet and join operations. We do the
same for the set of `maximal' such join-specifications, for a natural
definition of `maximal'. We also define functors from these lattices,
considered as categories, into a suitably defined category of frame-completions
of $P$, and construct right adjoints for them.
