Reflexivity and Order

Question

It seems to me that the most important concept that the idea of an order brings, is tied to the notion of assymetry ( or the weak antisymmetry, if you may ).

However, the way the order relations ( partial and strict ) are defined, we need to bring more to the table :

Transitivity and either Reflexivity or Irreflexivity.

For transitivity, even though it is not completely necessary for the concept of order ( i can think of many examples in the real-world, like some specific food chains ), i can understand how it can be important in Mathematics.

But I was just wondering why do we need to add :
. reflexivity ( $\forall x ( x R x ) $ ) or
. irreflexivity ( $\forall x \neg ( x R x)$ ).

Why are the Reflexivity axiom or the Irreflexivity axiom so important ?

Without those axioms, we would still include all the currently covered cases but would also let a relation like R = { (a,a), (a,b), (b,c),(a,c) } be an order relation.

I'm really a beginner in this whole deal of order, so excuse my possible ignorance.

Thanks in advance.

Answer 1

Antisymmetry and transitivity are the guarantors of order.

Reflexivity and irreflexivity provide the duality of partiality and strictness.

A partial order relation must be reflexive, antisymmetric, and transitive.   They behave analogously to $\leqslant$ , which is conceptually useful.

A strict order relation must be irreflexive, antisymmetric, and transitive.   They behave analogously to $<$ , which is conceptually useful.

You could define an antisymmetric and transitive relationship which is neither reflexive nor irreflexive.   It could be considered an order relation; but it will be neither partial nor strict.

Answer 2

You could define a relation with only anti-symmetry, or only anti-symmetry and transitivity, but having reflexivity(irreflexivity) is just... nice. For any total reflexive ordering $R$, you can define a related irreflexive order $R^*$(the order, sans $xRx \forall x$) or vice versa in which you can say things like if $aRb$ and $\neg(aR^*b)$, then $a=b$.

That being said, what you are talking about is just a more general relation. Language exists to talk about it(anti-symmetric transitive relations), but it isn't the most commonly talked about type of relation so it doesn't get a special name(like order).

Answer 3

The notion you are referring to is called a partial equivalence relation.

If you're asking why strict and partial orders are defined to require antireflexivity and reflexivity, well, the answer is just "that's the definition."

You may find that the notion of a partial equivalence relation captures your intuitive "essence" of what an "order" in the common sense on a set is, and thus may think that when we talk about an "order", we don't need reflexivity or antireflexivity, and we should be referring to partial equivalence relations when we use that language, but at that point we are just debating about what we should call things.

Partial equivalence relations, strict and partial orders, they are all just words. What matters is that all of these notions are clearly defined. The generalization you are referring to is defined, and you can talk about it.

If we didn't require strict and partial orders to have antireflexivity and reflexivity, then we would just have new names for those relations that did have those properties.