I am a bit confused about the role of propositional variables in the construction of the free Lindenbaum-Tarski algebra.
In the entry "Lindenbaum-Tarski algebra" on Wikipedia, in the section "operations", one can read:
"If the theory is propositional and its set of logical connectives is functionally complete, the Lindenbaum–Tarski algebra is the free Boolean algebra generated by the set of propositional variables."
In the entry "Lindenbaum-Tarski algebra" on PlanetMaths, one can read:
"It can be shown that the Lindenbaum-Tarski algebra of the propositional language L is a free Boolean algebra freely generated by the set of all elements $[p]$, where each $p$ is a propositional variable of L."
Which one is correct? Or, perhaps, are these statement both correct? Why?
The formulation in PlanetMath is attempting to guard against the possibility of non-logical axioms (i.e., non-trivial logical relationships amongst the propositional variables). Both statements are correct if the propositional theories don't have any non-logical axioms. However, for example, if you had an axiom saying that $p \Leftrightarrow q$, for every pair of propositional variables $p$ and $q$, then the equivalence classes $[p]$ would all be equal and the PlanetMath statement would be true, but (assuming you have more than one variable) the Wikipedia statement would be false.
However, the PlanetMath statement is not true in general: there are systems of non-logical axioms that result in Lindenbaum-Tarski algebras that are not free. (Thanks to Alex Kruckman for pointing this out.)