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Puzzle 4 Responses

I thought I might bring together all of the examples of posets that people have proposed:

  1. #4 John Baez: the set real numbers \(\mathbb{R}\), with the usual notion of \(\le\)
  2. #5 Dan Schmidt: the set of pairs \( (x,y) \in \mathbb R^2 \) where \( a \leq b \) when \( a_x \leq b_x \) and \( a_y \leq b_y \).
  3. #15 Matthew Doty:
    • Java classes form a preordered set when ordered by inheritance.
    • Natural numbers \(\mathbb{N}\) form a traditional poset under the relation "\(x\) divides \(y\)", often denoted "\(x\ |\ y\)".
    • For two sets \(A,B \subseteq \mathbb{N} \), we say \(A\) is Turing reducible to \(B\) if there is an oracle machine with oracle tape containing \(B\) that can compute the characteristic of \(A\). This is denoted \(A \leq_T B\). Under \(\leq_T\), \(\mathcal{P}(\mathbb{N})\) is a preordered set.
  4. #20 Alex Ortiz: An example of posets from analysis. If a measure space \((X,\mu)\) is \(\sigma\)-finite, then its \(L^p\) spaces (the collection of measurable functions \(f\colon X\to\Bbb C\) such that \(\int_X|f|^p\,\rm{d}\mu<\infty\)) can be preordered by set inclusion. In general this is not a total order. However, consider a finite measure space \((X,\mu)\). As a consequence of Hölder's inequality, we can actually put a total order on the \(L^p\) spaces with inclusion: $$ L^q(X,\mu)\subseteq L^p(X,\mu)\qquad\text{if $1\le p\le q\le \infty$.} $$
  5. #21-30, 37, 39, 40 David Tanzer:

    • Any collection of sets, ordered by the inclusion relation:

      • The lattice of subspaces of a vector space, ordered by inclusion.
      • Any set of algebras of the same type, ordered by inclusion.
      • #67 John Baez We could even go all out and consider the collection of all sets, ordered by the inclusion relation! This collection is too big to be a set. But we can get around that in various ways, e.g. by considering it as a proper class, or using a universe of "small" sets, which itself is a "large" set.
        A closely related example is Ord, the class of all ordinals. Ordinals form a totally ordered class. Ord starts out like this:
        $$ 0, 1, 2, 3, \dots, \omega, \omega + 1, \omega + 2, \dots, \omega \cdot 2, \dots, \omega^2, \dots, \omega^3, \dots, \omega^\omega, \dots, \epsilon_0, \dots $$
        but it goes on a lot longer. In fact it goes on longer than anything!
    • A collection of individuals, ordered by the ancestor relation. (Assuming that X is trivially an ancestor of itself.)

    • Any set of propositions, ordered by the implication relation.
    • Any subset of a poset is a poset.
    • A Cartesian product of posets is a poset.
    • We should also pay respects to the trivial poset {}.
    • And every set, along with the identity ordering relation, is a poset.
    • A set of formulas, ordered by the sub-formula relation, is a poset.
  6. #31 Joel Sjögren: The reflexive-transitive closure of any relation is a poset. Given A, define B inductively by (i) B(x,x) (ii) if B(x, y) and A(y, z) then B(x, z).
  7. #44 Jonatan Bergquist: the group of states in a non-reversible chemical reaction
  8. #48 Bob Haugen: a dataflow in a dataflow architecture
  9. #51 Artur Grzesiak: Let \(P = \{ (a,b): a < b \land (a,b) \in \mathbb{R^2} \}\) then following relations are posets:
    1. \((a,b)R(c,d) \iff a \le c \land b \le d \)
    2. \(xRy \iff x \subseteq y \)
    Since this is applied course one interpretation could be that first number denotes start of some process and the second number its end. So in case of 1. a process is larger than another one if it started after another one started and ended after another one ended. While in case of 2. process is larger than another one if the latter started and ended when the first one was active.
  10. #61 Daniel Cellucci: the dependency graph of spacecraft failures, where \(\le\) would indicate "leads to" or "increases the likelihood of".
    • #70 John Baez Indeed, this kind of example is very important in applications! A PERT chart is a way of planning tasks, where the edges indicate dependencies. There's more to a PERT chart than a mere preorder, as Simon Willerton has explained. We may get into that later. However, any PERT chart gives rise to a preorder.
  11. #73 Andrew Ballinger: git commits appear to form a poset.

Comments

  • 1.

    Missed material flows.

    Comment Source:Missed material flows.
  • 2.

    Thanks, Daniel, for collecting these! I've considered making each puzzle have its own discussion, or even wiki page, but let's wait a while and see how things shape up. (I will delete this digression in a while.)

    Comment Source:Thanks, Daniel, for collecting these! I've considered making each puzzle have its own discussion, or even wiki page, but let's wait a while and see how things shape up. (I will delete this digression in a while.)
  • 3.

    If \(X\) and \(Y\) are sets with preorders, you can define a preorder on \(X \sqcup Y\) by \(a \leq b\) iff \(a \leq b\) in \(X\) or \(a \leq b\) in \(Y\). This should be the coproduct in the category of posets. This is like putting \(X\) and \(Y\) right next to each other.

    Another preorder on \(X \sqcup Y\) is given by taking the one above and adding in \(a \leq b\) whenever \(a \in X\) and \(b \in Y\). This is like putting \(Y\) on top of \(X\).

    Comment Source:If \\(X\\) and \\(Y\\) are sets with preorders, you can define a preorder on \\(X \sqcup Y\\) by \\(a \leq b\\) iff \\(a \leq b\\) in \\(X\\) or \\(a \leq b\\) in \\(Y\\). This should be the coproduct in the category of posets. This is like putting \\(X\\) and \\(Y\\) right next to each other. Another preorder on \\(X \sqcup Y\\) is given by taking the one above and adding in \\(a \leq b\\) whenever \\(a \in X\\) and \\(b \in Y\\). This is like putting \\(Y\\) on top of \\(X\\).
  • 4.

    @JohnBaez I was just thinking that a collaborative document or wiki page would make generating the list much easier!

    Comment Source:@JohnBaez I was just thinking that a collaborative document or wiki page would make generating the list much easier!
  • 5.
    edited March 2018

    The Azimuth Forum was designed to make it easy to build the Azimuth Wiki. You can link to a wiki page, as I just did, simply using double brackets like this: [[Azimuth Wiki]]. So sure: pages that lots of people want to add information to can be wiki pages.

    If you want to create wiki pages, please read How to first, and each time you create a new wiki page, start a discussion in the category Latest Changes on the forum, with the same title as the new wiki page.

    If you want me to do this, just to show how it's done, let me know.

    However, over here Frederick Eisele has suggested starting a new discussion for each puzzle, as we've done here. So, there's a bit of a decision to made regarding discussions on the Forum versus wiki pages. Discussions are easier for people to notice; wiki pages are better for storing large amounts of information.

    We can just try stuff and see what works best, then formalize things later. Note that you can view the source for any comment on the Forum, which allows you to easily copy chunks of it to the Wiki.

    Comment Source:The Azimuth Forum was designed to make it easy to build the [[Azimuth Wiki]]. You can link to a wiki page, as I just did, simply using double brackets like this: `[[Azimuth Wiki]]`. So sure: pages that lots of people want to add information to can be wiki pages. If you want to create wiki pages, please read [[How to]] first, and each time you create a new wiki page, start a discussion in the category Latest Changes on the forum, with the same title as the new wiki page. If you want me to do this, just to show how it's done, let me know. However, [over here](https://forum.azimuthproject.org/discussion/comment/16293/#Comment_16293) Frederick Eisele has suggested starting a new discussion for each puzzle, as we've done here. So, there's a bit of a decision to made regarding discussions on the Forum versus wiki pages. Discussions are easier for people to notice; wiki pages are better for storing large amounts of information. We can just try stuff and see what works best, then formalize things later. Note that you can view the source for any comment on the Forum, which allows you to easily copy chunks of it to the Wiki.
  • 6.

    It makes logical sense to have a separate discussion for each exercise. The downside is that there will be a lot more discussions, which could make it harder to navigate to the main discussions - especially the lectures.

    We could control this using categories. All these discussions are already in the category Applied Category Theory Course.

    If you haven't already tried it, I suggest clicking on this category (one of the first categories, on the panel to the left). That immediately filters out, for example, the introductions, which are in category chat.

    My suggestion is to make another category Applied Category Theory Exercises, for the exercises.

    What do people think of this?

    Comment Source: It makes logical sense to have a separate discussion for each exercise. The downside is that there will be a lot more discussions, which could make it harder to navigate to the main discussions - especially the lectures. We could control this using categories. All these discussions are already in the category Applied Category Theory Course. If you haven't already tried it, I suggest clicking on this category (one of the first categories, on the panel to the left). That immediately filters out, for example, the introductions, which are in category chat. My suggestion is to make another category Applied Category Theory Exercises, for the exercises. What do people think of this?
  • 7.
    edited April 2018

    BTW I've been with the Azimuth Project for some time now, and am running the server for the wiki and the forum. I'm not from the first generation of Azimuth, though, which was ending as I joined. This third generation is very exciting!

    If people have suggestions about how to improve the structure of information here, that would be welcome.

    One thing that could be useful would be a wiki page which gives an organized index into the course -- with links to the lectures, the exercises, and what have you.

    Anyone interested in taking this on?

    If it gets into shape, I could look into adding a tab on the forum which points to this page.

    p.s. Just to let people know, I'm not really a professional admin, and I don't have that much spare time, so we'll have to balance out the desire to improve things with my availability. The more we can fit things into the standard framework, the more likely I will be able to make adjustments. If anyone is so inclined, it could be helpful if some people did research on open source Vanilla and its capabilities (this is our software here).

    Comment Source:BTW I've been with the Azimuth Project for some time now, and am running the server for the wiki and the forum. I'm not from the first generation of Azimuth, though, which was ending as I joined. This third generation is very exciting! If people have suggestions about how to improve the structure of information here, that would be welcome. One thing that could be useful would be a wiki page which gives an organized index into the course -- with links to the lectures, the exercises, and what have you. Anyone interested in taking this on? If it gets into shape, I could look into adding a tab on the forum which points to this page. p.s. Just to let people know, I'm not really a professional admin, and I don't have that much spare time, so we'll have to balance out the desire to improve things with my availability. The more we can fit things into the standard framework, the more likely I will be able to make adjustments. If anyone is so inclined, it could be helpful if some people did research on open source Vanilla and its capabilities (this is our software here).
  • 8.

    @DavidTanzer I think the place to start is with the Guide http://www.azimuthproject.org/azimuth/show/Forum+-+Guide I have also started the TOC that you suggested. http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory

    Comment Source:@DavidTanzer I think the place to start is with the Guide http://www.azimuthproject.org/azimuth/show/Forum+-+Guide I have also started the TOC that you suggested. http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory
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