MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

4 added 11 characters in body

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 $\phi \leq \psi$:

$$T \vdash \phi \leftrightarrow \phi\land rightarrow \psi$$

EDIT: And a simplification.

3 added 64 characters in body

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

$$\vdash T \vdash \phi \leftrightarrow \phi\land \psi$$

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 turning things aroundrelaxing the conditions:

Objects $\phi$ modulo:

$$\vdash \phi \leftrightarrow \phi'$$

Relation $\phi \leq \psi$:

$$T \vdash \phi \leftrightarrow \phi\land \psi$$

2 added 27 characters in body

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 $\phi \implies leq \psi$ iff

$$\vdash \phi \leftrightarrow \phi\land \psi$$

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 turning things around:

Objects $\phi$ modulo:

$$\vdash \phi \leftrightarrow \phi'$$

Relations:

Relation $\phi \leq \psi$:

$$T \vdash \phi \leftrightarrow \phi\land \psi$$