Posts tagged with partial ordering


  • 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.


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.



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+.