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,..nā1 $
I didn't get this one as well. Can anyone please provide some examples so that it is easier for me to visualize.