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As is well known, the axioms for a metric are:

  1. Positivity: $d(x,y)\ge 0$ and $d(x,y)=0$ iff $x=y$.

  2. Symmetry: $d(x,y)=d(y,x)$

  3. Triangle inequality: $d(x,z) \le d(x,y) + d(y,z)$

Now I notice that the second and third axiom can be replaced by a single axiom, let's call it the "thirds distance inequality" (the reason for this name will be obvious later):

Thirds distance inequality: $d(x,y) \le d(x,z) + d(y,z)$

It is immediately obvious that the thirds distance inequality can be derived from symmetry and triangle inequality. For the reverse direction, we need to use the property that $d(x,x)=0$.

Now, this fact reminded me of another fact, about the definition of equivalence relations. The usual axioms are:

  1. Reflexivity: $a=a$.

  2. Symmetry: $a=b \iff b=a$.

  3. Transitivity: $a=b \land b=c \implies a=c$.

Now symmetry and transitivity can be replaced by a single axiom, whose English name I don't know, but the German name is "Drittengleichheit", which I'd translate as "thirds equality":

Thirds equality: $a=c \land b=c \implies a=b$.

And again, you can prove thirds equality from symmetry and transitivity, but to prove symmetry and transitivity from thirds equality, you need to use reflexivity.

There is a striking similarity between the distance and the equivalence case: In both cases, we have the following ingredients:

  • Some entity applied to two objects (metric, equivalence relation)

  • An axiom that says something about applying the entity to an object and itself ($d(a,a)$, $a=a$)

  • An axiom stating symmetry.

  • An axiom relating the entity applied to $(a,c)$ to a combination of the entity applied to $(a,b)$ and $(b,c)$. This involves some combination operation ($+$, $\land$) and an operation relating the two sides ($\le$, $\implies$).

  • A replacement axiom for the latter two that just reverses one of the two combined entities by its symmetric reverse.

  • In both cases, you can derive the replacement axiom directly from symmetry and the original third axiom, but to do the reverse, you need the first, reflexive axiom.

So there seems to be a common pattern here. Is there a way to formally describe that pattern, and are there other cases where it also applies?

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    have a look at (https://unapologetic.wordpress.com/2010/03/24/measure-as-metric/) – 2017-02-12
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    @JeanMarie: Thanks for that link. However while it is certainly interesting, I'm not sure how exactly it relates to my question. – 2017-02-12
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    I think that your question (existence of a "bridge" between distance and equivalence axioms) could be asked in the same way between the axioms of a measure and the equivalence axioms. – 2017-02-12

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