Does there exist one relation is both reflexive, symmetric, transitive, antisymmetric

Question

Basically, we have 4 types of relations: reflexive, symmetric, antisymmetric, transtive. And then we separate 4 above types into 2 new definition:

1. One relation is reflexive, symmetric, transitve called equivalent relation.

2. One relation is reflexive, antisymmetrc, transitive called order relation.

All of them above are basic knowledge of elementary set theory.

So the question I wonder is "Does there exist one relation is both reflexive, symmetric, antisymmetric and transitive? If yes, so what is it called?"

Honestly, I have been finding out in the internet about my wonder, but of course I cannot see anything. Therefore, I post my question on here to ask everyone my question.
Thanks for your helping.

Answer 1

Suppose $\sim$ is symmetric. Then for all elements $a,b$ in the ambient set $S$, we have that $a\sim b$ implies $b\sim a$. But if $\sim$ is antisymmetric, then $a\sim b$ and $b\sim a$ together imply $a=b$. Hence $\sim$ is in fact equality.