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:

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

  • $\begingroup$ If $R$ is reflexive, then $x=y$ satisfies the conditions. Namely, if $xRz$ is true, then $xRx$ and $xRz$ are true. Perhaps you want $y\not=x,z$. $\endgroup$ Mar 2, 2017 at 1:39
  • $\begingroup$ 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. $\endgroup$
    – tokumei
    Mar 2, 2017 at 1:54

1 Answer 1


Short Answer: No.

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 $a < c < b$.

  • $\begingroup$ @NoahSchweber: It feels like it should be called that. Do you know if it's actually canon though? $\endgroup$ Mar 2, 2017 at 2:01
  • $\begingroup$ No idea, but I have seen it called that and I've never seen it called anything else, so . . . at least it's some kind of artillery? $\endgroup$ Mar 2, 2017 at 2:05

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