Why does a partially ordered set need to be reflexive?

Question

I understand that why a partially ordered set needs to be antisymmetric and transitive. I just can't see the logic behind why it has to be reflexive?

Answer 1

Because it mimics the usual meaning of "smaller-or-equal-than". In fact, there is a "dual" notion of "strict partial order":

A binary relation $R$ on a set $X$ is a strict partial order if and only if:

1. $\forall x\in X,\ \neg xR x$
2. $\forall x,y\in X,\ (xRy\land yRx)\to x=y$
3. $\forall x,y,z\in X,\ (xR y\land yR z)\to xR z$

Notice that $[(1)\land (2)]$ could have been restated equivalently as $[\forall x,y\in X,\ (\neg xRy\lor \neg yRx)]$.

An instance of these is the usual "$<$" on $\Bbb R$. Just as the names and examples suggest:

• for any partial order $P$ on a set $X$ the relation $x\Bbb S_Py\stackrel{\text{def}}\iff (x Py\land x\ne y)$ is a strict partial order;

• for any strict partial order $S$ on a set $X$ the relation $x\Bbb P_Sy\stackrel{\text{def}}\iff (x Sy\lor x= y)$ is a partial order;

• for any partial order $P$ and for any strict partial order $S$, $\Bbb S_{\Bbb P_S}=S$ and $\Bbb P_{\Bbb S_P}=P$.