Definition of the equality refered by the definition of a partial order

Question

A partial order $≤$ on a set $P$ must satisfy the following properties:

1. Reflexivity: For all $a \in P$, $a ≤ a$

2. Anti-symmetry: For all $a,b \in P$, $a ≤ b \land b ≤ a → a = b$

3. Transitivity: For all $a,b,c \in P$, $a ≤ b \land b ≤ c → a ≤ c$

But regarding the anti-symmtery, what equivalence relation does $=$ mean? Does it have to be the finest equivalence relation on $P$, or can it be coarser? Say, if $P$ is the set of triangles on the Euclidean plane $\mathbb{R}^2$:

• Usually, $=$ denotes the equality of the area of the triangles.

• There is also $≡$, which denotes the congruence of the triangles. This is also an equivalence relation.

• On the extreme, triangles could be considered equivalent only if the have exact same set of coordinates of vertices.

It is practical to define order over the $=$. It would sort triangles on their area. Defining order over the extreme equivalence would make lexicographical ordering of triangles. And I don't see a practical order relation over the $≡$.

So which equivalence relation does a partial order refer?

Answer 1

$a=b$ means $a$ and $b$ are the same element of $P$. If you want to put a partial order on the set of equivalence classes of $P$ (under some other equivalence relation) then you can do that. Then anti-symmetry has the same interpretation but in the set of equivalence classes.