- 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 japanese ⪰ english) 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 f ⪰j nor j ⪰ f … nor i ⪰ f nor f ⪰ 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.