Can a partial order be symmetric aside from being reflexive, antisymmetric, and transitive by definition?

Question

Can a partial order by symmetric in addition to being reflexive, antisymmetric, and transitive?

Also, can an equivalence relation be antisymmetric aside from being reflexive, symmetric, and transitive?

All of the definitions I see only state that a relation has to be those things in order for it to be considered a partial order or an equivalence relation. The definitions do not state that it has to be NOT antisymmetric or NOT symmetric.

Answer 1

The only reflexive, symmetric, and antisymmetric relation on a set $X$ is $\{(x,x):x\in X\}$. Reason: If $(x,y)$ is in the relation, then by symmetry so is $(y,x)$, then by antisymmetry $x=y$. This shows that the relation is contained in $\{(x,x):x\in X\}$, and the other containment is the definition of reflexivity.

Answer 2

The only way for a partially ordered set to be symmetric is if no two distinct elements $x$ and $y$ are comparable. Otherwise, if there is some $x<y$, then by symmetry $y<x$. Transitivity then implies $x<x$ which contradicts the fact that partial orders are irreflexive.

Edit: I was using a slightly different definition of partial order. (Irreflexive and transitive.)

Answer 3

Yes, although a poset whose partial ordering is symmetric will have a 'trivial' partial ordering, namely $a \leq b \Rightarrow a=b$