Interior algebra

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Description:
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

&lang;S, ·, +, ', 0, 1, <sup>I</sup>&rang;


where

&lang;S, ·, +, ', 0, 1&rang;


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

  1. x<sup>I</sup> ≤ x
  2. x<sup>II</sup> = x<sup>I</sup>
  3. (xy)<sup>I</sup> = x<sup>I</sup>y<sup>I</sup>
  4. 1<sup>I</sup> = 1


x<sup>I</sup> is called the interior of x.

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

  1. x<sup>C</sup> ≥ x
  2. x<sup>CC</sup> = x<sup>C</sup>
  3. (x + y)<sup>C</sup> = x<sup>C</sup> + y<sup>C</sup>
  4. 0<sup>C</sup> = 0


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