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: ari≻shem and shem≻zvi implied ari≻zvi)
And, by definition, even I was more popular than the nullset. (thanks, mathematics)
