A closure algebra is a Boolean algebra with operator, , which satisfies: for all , .
In general, if is a closure algebra and , we say that is closed if and open if , where is the dual operator of .
A closure algebra is sometimes written in terms of instead of and is then called an interior algebra.
If is a closure algebra, let be the set of open elements in , then has the natural structure of a Heyting algebra. Moreover any Heyting algebra can be represented as the algebra of open elements of a closure algebra.
Closure algebras underly the algebraic semantic models of the epistemic logic .
The algebraic semantics of uses polyclosure algebra?s. Here there are many different closure operators on the Boolean algebra.