# The opposite of Transitivity (relations, set theory)

A relation $R$ is called 'transitive' when it upholds

$$xRy ~\text{ and }~ yRz \implies xRz.$$

Is there a name for a relation that upholds the somewhat opposite property:

$$xRz \implies \exists y ~\text{ such that }~ xRy \text{ and } yRz \enspace ?$$

• If $R$ is reflexive, then $x=y$ satisfies the conditions. Namely, if $xRz$ is true, then $xRx$ and $xRz$ are true. Perhaps you want $y\not=x,z$. Mar 2, 2017 at 1:39
• Adding some background: I'm thinking about translation. Due to ambiguity it is not transitive, but the nameless property above seems to be correct -- if x translates to z (in languages L1 and L3), there should be an instance y in (any) language L2 such that x translates to y and y translates to z. Mar 2, 2017 at 1:54

Long Answer: So, I wouldn't exactly call this the opposite of the transitivity relation. Surely, you can have both properties hold. If I were going to give this property a name, one might call it a density property. If $R$ is the ordering on $\mathbb{Q}$, this is property says that is $a < b$ then $\exists c$ such that $a < c < b$.