Posts tagged with poset

Paul Finsler believed that sets could be viewed as generalised numbers. Generalised numbers, like numbers, have finitely many predecessors. Numbers having the same predecessors are identical.

We can obtain a directed graph for each generalised number by taking the generalised numbers as points and directing an edge from a generalised number toward each of its immediate predecessors.

It has been shown that these generalised numbers can be “added” and “multiplied” in a natural way by combining the associated graphs. The sum a+b is obtained by “hanging” the diagram of b onto that of a so the bottom point of a coincides with the top point of b. The product a·b is obtained by replacing each edge of the graph of a with the graph of b where the graphs are similarly oriented.

Paul Finsler, David Booth, Renatus Ziegler in Finsler set theory: platonism and circularity




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  • I like Indonesian food better than Japanese food i ⪰ j, and
  • I like Japanese food better than English food j ⪰ e.
  • I also like French food better than English food f ⪰ e, but
  • I see French food as so different from the “exotic Eastern” foods that I can’t really say whether I prefer French food to Indonesian f≹i or Japanese f≹j.

    I would just be in a different mood if I wanted French food than if I wanted “exotic Eastern” food.

So my restaurant preferences are shaped like a poset. In a poset some things are comparable  and some things ain’t . Popularity is shaped as a poset and so is sexiness. Taste in movies is a poset too. The blood types have the same mathematical form as a poset but only if you reinterpret the relation  as “can donate to” rather than “is better than”. So not really the same as ethnic food.

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Partial rankings | orders are transitive, so

  • (indonesian ⪰ japanese and japaneseenglish) implies indonesian ⪰ english.

That means I can use the “I prefer " symbol to codify what I said at the outset:

  • Indonesian  Japanese English
  • French English
  • neither⪰j nor⪰ f … nor⪰ f nor⪰ j (no comparison possible )


Posets correspond nicely to graphs since posets are multitrees.

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Total orders — where any two things can be ranked  — also correspond to graphs, but the edges always line up the nodes into a one-dimensional path. So their graphs look less interesting and display less weird dimensional behaviour. Multitrees (posets) can have fractional numbers of dimensions, like 1.3 dimensions. That’s not really surprising since there are so many kinds of food / movies / attractiveness, and you probably haven’t spent the mental effort to precisely figure out what you think about how you rate all of them.

Rankings | orders are a nice way to say something mathematical without having to use traditional numbers.

I don’t need to score Indonesian at 95 and score Japanese at 85. Scores generated that way don’t mean as much as Zagat and US News & World Report would like you to think, anyway — certainly they don’t have all the properties that the numbers 85 and 95 have.

It’s more honest to just say Indonesian ⪰ Japanese lexicographically, and quantify no more.







At my secondary school, the high-scoring wide receiver was more popular than the fat lineman. And the fat lineman was more popular than the team statistician. But you couldn’t really compare the wide receiver’s popularity to that of the actor who got most of the lead roles. They were admired in different circles, to different degrees, by different people. With so little overlap, a hierarchy must treat them as separate rather than comparable.

So popularity = a partial order (and, possibly, an inverted arborescence or join-semilattice). Sometimes there is a binary relation  between two people such that one is-more-popular than the other. Sometimes you just can’t say. And no such relation exists. (neither geoff ≻ ian nor ian ≻ geoff)

Transitivity did hold at my school, so if you were more popular than geoff, you were by extension more popular than anyone than whom geoff was more popular. ( ari, shem, zvi: arishem and shemzvi implied arizvi)

And, by definition, even I was more popular than the nullset. (thanks, mathematics)




Blood types form a topological space (and a complete distributive lattice). There are three generators: A, B, and Rh+.

Above the “zero element” is the universal donor O− and the “unit element” is the universal receiver AB+.

A topological space contains a zero object, maybe other objects, and all unions & intersections  of anything in the space.  So taking the power set  of {A, B, +} yields the “power set topology” which I drew above. AB+ is the 1 object and “nullset” O− is the 0 object.

A lattice has joins  & meets  which function like  and  in a topological space. Like 1 or True in a Heyting algebra, blood type as a power-set topology has one “master” object AB+.




One example of a total ordering is the “hotness” scale from 110.

Because of the widespread disagreement about the meaning of the numbers, the only thing one can infer based on a man’s rating of a woman is that she is more attractive than those who score below her.

YOUTH

The Hotness Scale derives, I think, from a need to explain one’s tastes to peers and hear them justified.

It typically surfaces in sleepover conversations like this:

  • Chris (secretly likes Kelly Russell): Who do you think is hotter: Liz Jones, or Kelly Russell?
  • Dave: Are you kidding?! Liz Jones is waaaay hotter.
  • Chris: Oh, yeah. I mean, obviously. I was just checking. I just meant, you know, that I think Kelly Russell is like maybe a 7.
  • Dave: Are you crazy?! She’s like a 2.
  • Chris: Come on, 2 is like people who have skin grafts. 2 is people who were burned in fires.
  • Dave: Whatever. Maybe.
    (sheepish retreat to celebrity hotness)
    But man, Jane March is the hottest woman on the planet.
  • Chris: No way, Jenny McCarthy is hotter.

I’m a little embarrassed to admit (though I’ll still admit it) that when my best friend and I started using the hotness scale, we scored girls in different categories, like

  1. tan
  2. boobs
  3. personality
  4. legs
  5. I forget what else
  6. overall score

Yeah, we were really cool. (Also we were really twelve.)



DISPUTES

Fast forward to college. We guys were joking about using the ten-point scale, which by then was passé (although I did once use the phrase “a Bloomington 6 is a hometown 9”). We were trying to answer, what is the difference between a 6 and a 7 anyway? And is the distance between 6 and 7 greater or less than the distance between a 9 and a 10?

Everybody had taken calculus by this point so statements involving derivatives were bandied about (even though none of us meant to use real numbers … it was calculus as metaphor).

One guy proposed that each number should correspond to a decile —

  • 1 to the ugliest decile,
  • 10 to the hottest decile,
  • and so on.

 Someone else said that one’s initial reaction put a girl either in

  • the >4 (most of the time) or
  • <5 — but that since no one would ever hit on someone in the latter category,
  • in fact 1≈2≈3≈4.

Another said that he never assigned a 10 to anybody because that would mean he had met his wife. Um, yeah … we were still super cool.

Also contentious was whether each of us accepted the truth of whatever our own numerical ranking was. All I know is that whatever I said my score was, I secretly hoped it was 2 points higher.

There was a lot of inconsistency to the scores, which is why I’m bringing this up under the topic of rank without distance measure. Although I would wager that transitivity is violated, so perhaps this scale does not have a rational basis.



PEOPLE CAN BE ATTRACTIVE IN DIFFERENT WAYS

As I’m writing all of this I desperately want to jump ahead to partial orderings. But I haven’t defined them yet and I refuse to link to Wikipedia, so I’ll have to put that topic off.

Suffice to say that attraction is a perfect jumping-off point for one further generalization I want to make in order to get mathematics into bed with human experience.

Not everyone can be ranked side-by-side against everyone else. People can be attractive for different reasons (multiple >'s) and some people you just aren't comparable. All of these reasons and more are great justifications for switching to posets.



POSTSCRIPTA

I’d be interested to hear other interpretations of the hotness scale, or other scales of attractiveness, and how they evolved over time. What measure, if any, do you give guys when you’re in your 30’s or 40’s?

PS When I was 24, my girlfriend told me that “24 is a very hot age”. Ha ha.

PPS Tim Ferriss claims to be able to quantify the difference between a 6 and a 9. Tell that to Jimi Hendrix.