Editing Lindenbaum–Tarski algebra (section)

== Operations ==

The operations in a Lindenbaum–Tarski algebra ''A'' are inherited from those in the underlying theory ''T''. These typically include [[logical conjunction|conjunction]] and [[disjunction]], which are [[well-defined]] on the equivalence classes. When [[negation]] is also present in ''T'', then ''A'' is a [[Boolean algebra (structure)|Boolean algebra]], provided the logic is [[classical logic|classical]]. If the theory ''T'' consists of the [[propositional calculus|propositional tautologies]], the Lindenbaum–Tarski algebra is the [[free Boolean algebra]] generated by the [[propositional variable]]s.

If ''T'' is closed for deduction, then the embedding of ''T/~'' in ''A'' is a [[Filter (mathematics)|filter]]. Moreover, an [[ultrafilter]] in A corresponds to a complete consistent theory, establishing the equivalence between [[Lindenbaum's lemma|Lindenbaum's Lemma]] and the [[Ultrafilter on a set|Ultrafilter Lemma]].