2018/06/02 by Egrot, Rob
#06A15 #06D22 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1806.00642
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.