In Lattice Theory p. 30 3rd edition:
Lemma 3. In any distributive lattice, every polynomial is equivalent to a join of meets, and dually: $p(x_1,...,x_r)=\lor_{\alpha \in A}\{ \land_{S_\alpha} x_i \}= \land_{\delta \in D} \{ \lor_{T_\delta} x_j \}$
where $S_\alpha$, $T_\delta$ are nonempty sets of integers.
So far I understand this notation and thought I had proved the lemma pretty simply myself, but the first line of Birkhoff's proof threw me. Here it is:
Proof : Each $x_i $ can be so written where $A$ or $D$ is the family of sets consisting of the single one element set $x_i$....
If A is the "family of sets consisting of the one element set $x_i$ "
Would A look like: $A= \{ \{ x_i\} \}$ or $ A= \{ \{ x_i \} ,\{ x_i, x_j\},\{ x_i, x_k\}, \{ x_i,x_j,x_k \}\}$
And then an example of an $S_\alpha$ would be for instance $S_{\{ x_i\} }$? And what would be the members of this set?
I think this place in the book is written not very carefully. $A$ is the family of indices and in this case $A=\{\alpha\}$, the one-element family, and $S_\alpha=\{x_i\}$.