Interior algebra

In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebra are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.

Definition

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. x^I ≤ x
2. x^II = x^I
3. (xy)^I = x^Iy^I
4. 1^I = 1

x^I is called the interior of x.

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

1. x^C ≥ x
2. x^CC = x^C
3. (x + y)^C = x^C + y^C
4. 0^C = 0

If the closure operator is taken as primitive, the interior operator can be defined as...
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