Interior Algebra

An interior algebra is an algebraic structure with the signature

〈S, ·, +, ', 0, 1, I〉

where

〈S, ·, +, ', 0, 1〉

is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities:

1. xI ≤ x
2. xII = xI
3. (xy)I = xIyI
4. 1I = 1

xI is called the interior of x.

The dual of the interior operator is the closure operator C defined by xC = ((x ' )I )'. xC is called the closure of x. By the principle of duality, the closure operator satisfies the identities:

1. xC ≥ x
2. xCC = xC
3. (x + y)C = xC + yC
4. 0C = 0

If the closure operator is taken as primitive, the interior operator can be defined as xI = ((x ' )C )'. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form 〈S, ·, +, ', 0, 1, C〉, where 〈S, ·, +, ', 0, 1〉 is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras form dual pairs, and are paradigmatic instances of "Boolean algebras with operators." The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm.