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- W2807352432 abstract "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." @default.
- W2807352432 created "2018-06-13" @default.
- W2807352432 creator A5000606764 @default.
- W2807352432 date "2018-06-02" @default.
- W2807352432 modified "2023-09-27" @default.
- W2807352432 title "Categories of frame-completions and join-specifications" @default.
- W2807352432 hasPublicationYear "2018" @default.
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