Consider the set $S = \left\{ a , b , c , d \right\}$ . Consider the following 4 partitions $\pi_1,\pi_2,\pi_3,\pi_4$ on $S : \pi_1 = \left\{\overline{abcd}\right\} , \pi_2 = \left\{\overline{ab}, \overline{cd}\right\}, \pi_3 = \left\{\overline{abc}, \overline{d}\right\}, \pi_4 = \left\{\bar{a}, \bar{b}, \bar{c}, \bar{d}\right\}$. Let $\prec$ be the partial order on the set of partitions $S' = \{\pi_1,\pi_2,\pi_3,\pi_4\}$ defined as follows: $ \pi_i \prec \pi_j$ if and only if $\pi_i \text{ refines }\pi_j $. The poset diagram for $ (S',\prec)$ is:
Reveal answer
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$\pi_1$= {$\overline{abcd}$} refines
$\pi_2$= {$\overline{ab}$,$\overline{cd}$}
$\pi_3$= {$\overline{abc}$,$\overline{d}$}
and these both refines
$\pi_4$={$\bar{a}\bar{b}\bar{c}\bar{d}$}