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I'ts know that given a partition of $E$ we can construct an equivalence relation on $E$. (a subset of $E^2$)

My question: if we had a partition of $E^2$ how do we define an equivalence relation on $E$. (a subset of $E^4$)

TWJ
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  • Why would a partition of $E^2$ give an equivalence relation on $E$? Why would $E$ be a subset of $E^4$? Or did you mean $E^2$ when you wrote $E$? And if so why not just let $F=E^2$ reducing to a problem already solved? – Gerry Myerson May 20 '19 at 04:41
  • @GerryMyerson I'm trying to deal with the problem that not already solved – TWJ May 20 '19 at 04:51
  • And I'm trying to understand why you are writing that an equivalence relation on $E$ is a subset of $E^4$. – Gerry Myerson May 20 '19 at 04:53
  • by detention a binary relation on $E$ is a subsets of $E^2$. Let's call this one quadrany relation ? – TWJ May 20 '19 at 04:57
  • OK, a partition of $E^2$ defines a subset of $E^4$ in exactly the same way that a partition of $E$ defines a subset of $E^2$. Just let $F=E^2$, reducing to a problem already solved. (detention? definition?) – Gerry Myerson May 20 '19 at 05:00
  • that would give me a relation over $F=E^2$. I want a relation over $E$ – TWJ May 20 '19 at 05:04
  • OK, you want a 4-place relation on $E$. Then: if $(a,b)$ and $(c,d)$ are in the same part of the partition of $E^2$, then $(a,b,c,d)$ is in the 4-place relation on $E$. – Gerry Myerson May 20 '19 at 05:22
  • Is that OK, TWJ? – Gerry Myerson May 21 '19 at 12:45
  • Are you still here, TWJ? – Gerry Myerson May 23 '19 at 00:26
  • sorry i didn't see your comment, we define a 2-place relation as follow $(a,b)$ are in the relation iff there an element of the partition that contains $a$ and $b$. how can we define it for the 4-place relation – TWJ May 23 '19 at 03:46
  • $(a,b,c,d)$ is in the 4-place relation if and only if there is an element of the partition that contains both $(a,b)$ and $(c,d)$. – Gerry Myerson May 23 '19 at 05:27
  • It would be difficult to define transitivity on that – TWJ May 23 '19 at 05:33
  • You're making me crazy. You want an equivalence relation on $E$. An equivalence relation on $E$ is a subset of $E^2$. But you want your equivalence relation to be a subset of $E^4$. Next, you'll be asking for a triangle with four sides, and a two-digit number exceeding $100$. – Gerry Myerson May 23 '19 at 10:20
  • in the literature the n-ary relations are known. a relation don't have to be binary, and a polygon don't need to be triangle – TWJ May 23 '19 at 19:51
  • If you are so on top of the literature, then why are you asking questions here? Does the literature say anything about subsets of $E^4$ as equivalence relations on $E$? – Gerry Myerson May 23 '19 at 22:48
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    no it dosen't. Now i see that my question dosen't make a lot of sens. i was hoping someone have a straightforward answer. i just realized that it don't exist – TWJ May 23 '19 at 23:04
  • Good. Maybe you could write up an explanation of why it doesn't exist, and post that as an answer. – Gerry Myerson May 23 '19 at 23:09
  • i don't think many people are interested in why it dosen't exist a triangle with 4 legs – TWJ May 24 '19 at 09:01

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