In Aumann's ``Aggreeing to disagree" classic paper, the author defines for different partitions of a set $\Omega$, that refers to the state of the word, the join $\vee$ and the meet $\wedge$. For any two partitions $\Pi_1$ and $\Pi_2$ of $\Omega$, the join is defined as $\Pi_{\vee}=\Pi_1\vee \Pi_2$ which is the coarsest common refinement of $\Pi_1$ and $\Pi_2$ and the meet is defined as $\Pi_{\wedge}=\Pi_1\wedge \Pi_2$ which is the finest common coarsening of $\Pi_1$ and $\Pi_2$. From lattice theory $\Pi_{\vee}$ denotes the least upper bound, nameley the smallest $\sigma-$algebra $\mathcal{G}$ such that $\Pi_1\subseteq \mathcal{G}$ and $\Pi_2\subseteq \mathcal{G}$ while $\Pi_{\wedge}$ denotes greatest lower bound, namely the largest $\sigma-$algebra $\mathcal{G}$ such that $\mathcal{G} \subseteq \Pi_1$ and $\mathcal{G} \subseteq \Pi_2$.
Could someone give an example of the coarest common refinement and the finest common coarsening and where could we see a diagramatic illustration? In a book for example?
By referring to the smallest and largest $\sigma-$algebra do we refer to the union and the intersection of the partitions $\Pi_1$ and $\Pi_2$, namely $\sigma(\Pi_1 \cup \Pi_2)$ and $\sigma(\Pi_1 \cap \Pi_2)$?
By referring to the smallest and largest $\sigma-$algebra do we mean that we refer to the those $\sigma-$algebras with the best and worst possible predictions in terms of conditional expectations $\mathbb{E}[\cdot |\sigma(\mathcal{G})]$