Down-sets are more convenient than up-sets for many purposes in order theory.

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• The order relation ≤ on a poset P is determined entirely by the down-sets in P.claim0.767
• Powersets are too nice for pure set models; ordered set models are richer and can capture more computational behaviors.claim0.758
• Above always manages to assert a claim of metaphysical superiority, as if it were of a higher order, while below takes the role of support, solid ground, the foundation and/or actuality of a thought form.claim0.734
  Spatial verticality inherently projects hierarchical meaning.
• Order provides terminology, notation, and models across diverse areas of computer science including comparisons of size, information content, and definedness.claim0.733
• It is this injection of definite, strong, geometrical order that allows the profound depth of the made thing — the building in the land — and it is from this that the order must and will arise.claim0.733
  The geometric injection is the causal source of the building's profound depth
• It is, indeed, this pattern of the middle-sized entities, which most strongly gives overall geometric order to the building. Without it, a building is very unlikely to be beautiful.claim0.732
  Identifies the pattern of middle-range entities as the primary source of overall geometric order and beauty
• It is more important to get the rooms right, one by one, than it is to have a coherent 'plan'.claim0.730
  A design principle that rejects plan-driven layout.
• Scale, above all, is vital; the right size of pieces only becomes clear when one is standing looking down at them.claim0.726
  Emphasizes the importance of full-scale physical judgment over scaled drawings.