Let $R$ be a binary relation. Consider the property:
$$ a R b \wedge b R c \implies \neg(c R a)$$
This looks a lot like transitivity:
$$ a R b \wedge b R c \implies a R c$$
In my head I've been calling it "weak transitivity", in that it restricts the existence of "loops" (i.e. $a R b \wedge b R c \wedge c R a$), but it allows more types of relations than transitivity.
I have figured that if the relation is also asymmetric, then this property and transitivity imply each other. But they are not equivalent in the general case. For example, let $P$ be the relation "is a parent of." $P$ satisfies the unnamed property, but is not transitive. if $a P b$ and $b P c$, then $c$ cannot be the parent of $a$, but also $a$ is not the parent of $c$ (they are an ancestor, but not a parent.)
So this property is not equivalent to transitivity. Does it have a name? Is it equivalent to the combination of other named properties?