Categories of frame-completions and join-specifications

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.