This is the following excerpt from my book:
"To each nonempty proper$^2$ order filter $A$ of the poset $(2^V ,\subseteq)$ (the set of all subsets of $V$ ordered by inclusion) associate a term formed as a join of meets, called a normal term$^3$. For each element of $A$, a nonempty set of variables since $A$ is proper, form the meet of those variables. Then form the join of all those meets, of which there is at least one since $A$ is nonempty."
where the footnotes are:
$2.$ When working with lattices with a constant $0$ denoting the least element we allow the empty filter as well. Independently, in the presence of a constant $1$ denoting the greatest element we also allow the improper filter.
$3.$ Some readers will recognize this as disjunctive normal form with the additional requirement that the set of disjunctions is closed under conjoining disjuncts with variables.
Question: I have come across DNF's in propositional logic. I have absolutely NO idea what the definition of "normal term" means here? Could someone illustrate with an example? For example, let $V = \{x , y, z \}$ and let $A = \{ \{x \}, \{x,y\}, \{x,z\} , \{x,y,z\} \}$ be a nonempty proper order filter. What is the normal term of $A$?
It looks like the second sentence (the one beginning "For each element of $A$...") actually gives the definition of how to construct the normal term of $A$. Just follow the directions, looks like.
So for the example you gave, the normal term of $A$ would be $(x)\vee(x\wedge y)\vee (x\wedge z)\vee (x \wedge y \wedge z)$.