Attached is an image 
I found on a joke Reddit post. In a subsequent conversation I had with a friend about the image he made a comment and I replied something to the effect "let's just assume it's a partial ordering".
Well. It's not a partial ordering for certain because it is not anti-symmetric. Visual Basic > FORTRAN and FORTRAN > Visual Basic.
Is it transitive however? JavaScript is > visual basic > FORTRAN but FORTRAN > visual basic. What's a good example of a non-transitive anti-symmetric relation? I feel like this is really basic stuff but it's been a long time since I've sat in a mathematical logic class.
Thanks!
The term for a "partial order" that's not anti-symmetric is a preorder. That is, a preorder is a reflexive, transitive binary relation. If you take the arrows literally, then it is not a preorder because it is not transitive, but the intent is that you take the transitive closure of the relations indicated by the arrows. That is to say, to explicitly represent the transitive closure, you should have an arrow from "Perl programmers" to "PHP programmers" for example, among many others, but the idea is that arrow is implied.
That said, it is still not a preorder because it is not reflexive. Often when we have such diagrams we also consider self-loops implied and thus would take the reflexive-transitive closure of the relation represented by the arrows, but in this case, the meaning of the relation suggests that it is not reflexive. That is, Perl programmers don't consider themselves superior to other Perl programmers, for example. (If you do want to take it as each group considers itself superior to itself, then the implied relation would be a preorder.) Some cases of this do happen, i.e. whenever there's a cycle, e.g. Assembler programmers consider themselves superior to Assembler programmers. If you want to disallow this, then we can't view the diagram as implicitly specifying a transitive relation.
I'm not really sure why you ask about a non-transitive anti-symmetric relation as you seem to believe (correctly) that the diagram represents a transitive non-anti-symmetric relation. At any rate, a somewhat "everyday" example would be a relation, $R$, on people such that for people $x$ and $y$, $R(x,y)$ if and only if $y = x$ or $y$ is a child of $x$. This is anti-symmetric since no one is a child of their children, so if $R(x,y)$ holds then $R(y,x)$ can only hold if $y=x$. It's not transitive because your children's children are not your children.