From Wikipedia I learn:
The Lindenbaum algebra A of a theory T consists of the equivalence classes of sentences of T. The operations in A are inherited from those in T.
If there are disjunction, conjunction and negation, A is a Boolean algebra and can be seen as a poset:
The objects of A are sentences $\phi$ modulo
$$T \vdash \phi \leftrightarrow \phi'$$
There is a relation $\phi \leq \psi$ iff
$$T \vdash \phi \leftrightarrow \phi\land rightarrow \psi$$
Lindenbaum algebras are a bit boring since — for example — all complete theories T have the same two-element Lindenbaum algebra.
They might be a bit more interesting when relaxing the conditions:
Objects $\phi$ modulo:
$$ \vdash \phi \leftrightarrow \phi'$$
Relation $\phi \leq \psi$:
$$T \vdash \phi \leftrightarrow \phi\land rightarrow \psi$$
I just want to know where I can learn more about this approach?
EDIT: I made two corrections due to Joel's answer.
EDIT: And a simplification.
