2
$\begingroup$

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:

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

$\endgroup$
2
  • $\begingroup$ 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

3
$\begingroup$

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 $a < c < b$.

$\endgroup$
2
  • $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.