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I was referring to this article here related to the formation of a complete lattice by the partitions of a set. The article has stated that the partitions not only form the lattice for themselves but also for the equivalence relations.

I know that each partition has a corresponding equivalence relation. However, I don't get the derivation here in this article.

Things like the following

$$ a \equiv b(modE) $$

I didn't get the things given in the remarks specially after it says

Correspondingly, the partition lattice of S also defines the lattice of equivalence relations $\Delta$ on S

Any guidance pls?

It mentions that

Given a family $\{E_i|i \in I\}$ of equvialence relations on S , we can explicitly describe the join E:=V $E_i$ of $E_i$ , as follows:

$a\equiv b$(modE) iff there is a finite sequence

$a=c_1,c_2,...c_n=b$ such that

$c_k \equiv c_{k+1}(modE_{i(k)})$ for $k=1,..n−1 $

I didn't get this one as well. Can anyone please provide some examples so that it is easier for me to visualize.

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    $a \equiv b(\text{mod}E)$ means that $a$ and $b$ lie in the same cell of $E$.2012-09-12
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    @Michael, we have cells only in the partition isn't it? So you mean in the same cell of the partition equivalent to the equivalence relation E isn't it?2012-09-12
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    Yes. Actually, in the present context of equivalence relations it simply means $aEb$.2012-09-12
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    @Michael, I have also added one more thing that I didn't understand in the original question. Can you give me some examples like with a set A and make it clear to me. I think if I visualize with examples it will be clear to me2012-09-12
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    Please don't change the question. I can explain these things to you on chat: http://chat.stackexchange.com/rooms/36/mathematics2012-09-12
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    @Michael sure thanks. I have just joined the chat may be we can talk there2012-09-12

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