In the book “A Functorial Model Theory” by Nourani (pg152), it is stated that
However, I didn’t understand what does he mean? Because a complete lattice is not even necessarily distributive whereas we define Boolean algebras on distributive lattices. What does he mean by saying “Any complete lattice is a Boolean algebra”? Is this statement true?
Consider the complete Boolean algebra with four elements. Let the elements be $0$, $a$, $b$, and $1$ where $0 \leq a, b \leq 1$. The elements $a$ and $b$ satisfy $a \wedge b = 0$ and $a \vee b = 1$. Add a third element $c$ such that $0 < c < 1$, $0 = a \wedge c = b \wedge c$, and $1 = a \vee c = b \vee c$. This five element lattice is not distributive. Notice that $(a \wedge b) \vee c = 0 \vee c = c$ but $(a \vee c) \wedge (b \vee c) = 1 \wedge 1 = 1$.