Question about partial order

As I understand it, for set S a partial order R is any subset of S X S
having reflexive, antisymmetric, and transitive properties. If I is
the set of all integers, then it is said that divisibility does not
define a partial ordering of I because it is not antisymmetric, that is
2 divides -2 and -2 divides 2, but 2 is not equal to -2. However if

R = {(n,n) : all n in I}

then we can have divisibility defining a partial order, since this set
satifies antisymmetry and transitivity as well, albeit vacuously.
So, for any relation such as divisibility, does it either define an
ordering or not in general, or are there instances of each case for a
given relation?

All help appreciated.

.