Posts tagged with transitivity

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The word < is normally defined to mean less than in some quantifiable sense. For example, considering the set {3,6,1441}, one could say that 3<6<1441.

But in the abstract language of partially ordered sets, < is reinterpreted many ways — to mean proper subset of ⊂  (contained by), divides, “is hotter than" or … any transitive relation — even begat.

Consider the set

  • {Cain, Enoch, Irad, Mehujael, Methusael, Lamech₁, Jabal, Jubal, Tubal-cain, Naamah} ∪ {Adam, Abel, Seth} ∪ {Seth, Enosh, Kenan, Mahalalel, Jared, Enoch₂, Methusaleh, Lamech₂, Noah} ∪ {Noah, Shem, Ham, Japheth}.

Transitivity means that it’s impossible for Adam < … < Adam < … (where Adam refers to the same man, not to another person also named “Adam”. We can call him Adam₀ if it’s a problem).

Then the fourth and fifth chapters of Genesis yield the following relations among the members of that set.

  • Cain < Enoch₁ < Irad < Mehujael < Methusael < Lamech₁ < Jubal
  • Lamech₁ < Jabal
  • Lamech₁ < Tubal-cain
  • Lamech₁ < Naamah
  • Adam < Cain
  • Adam < Abel
  • Adam < Seth
  • Adam < Seth < Enosh < Kenan < Mahalalel < Jared < Enoch₂ < Methuselah < Lamech₂ < Noah
  • Noah < Shem
  • Noah < Ham
  • Noah < Japheth

It’s not like we reduce Enoch₂ to “the thing between Jared and Methuselah” — there is other information attached to Enoch₂ such as that he walked with God and was no more (whereas all the others were noted to have died). Likewise to say that 18 is the integer between 17 and 19 isn’t to ignore the fact that 6 divides 18 or that it represents a legal bright-line in some countries.

Nor do we assume that every pair (a,b) from that set should be comparable. (In a totally ordered set either a<b or b<a, ∀a,b.) But in case of the begats:

  • Mehujael Noah
  • Shem Tubal-cain
  • Jared Jabal

You can put more than one order on a set. For example, regarding salsas from the set {mild, medium, hot}:

  • mild < medium < hot, where < means the predicate “is less spicy than”
  • medium hot mild, where x≻y means the predicate “She prefers x to y”.

One could launch off into defining ≻ᵢ's for every  of the population. (This is the kind of thing one needs to do in talking about Arrow’s Theorem.) That’s when notation begins to suck.