5 Matching Annotations
  1. Jun 2018
    1. The preservation of meets and joins, and hence whether a monotone map sustainsgenerative effects, is tightly related to the concept of a Galois connection, or moregenerally an adjunction.
    2. In his work on generative effects, Adam restricts his attention to maps that preservemeets, even while they do not preserve joins. The preservation of meets implies that themapbehaves well when restricting to a subsystem, even if it can throw up surpriseswhen joining systems
    3. n [Ada17], Adam thinks of monotone maps as observations. A monotone map:P!Qis a phenomenon ofPas observed byQ. He defines generative effects of such a mapto be its failure to preserve joins (or more generally, for categories, its failure topreserve colimits)
    4. Example1.61.Consider the two-element setPfp;q;rgwith the discrete ordering.The setAfp;qgdoes not have a join inPbecause ifxwas a join, we would needpxandqx, and there is no such elementx.Example1.62.In any posetP, we havep_pp^pp.Example1.63.In a power set, the meet of a collection of subsets is their intersection,while the join is their union. This justifies the terminology.Example1.64.In a total order, the meet of a set is its infimum, while the join of a set isits supremum.Exercise1.65.Recall the division ordering onNfrom Example 1.29: we say thatnmifndivides perfectly intom. What is the meet of two numbers in this poset? Whatabout the join?

      These are all great examples. I htink 1.65 is gcd and lcm.

  2. Oct 2016