Editing Interior algebra (section)

=== Derivative algebras ===
Given an interior algebra '''''A''''', the closure operator obeys the axioms of the [[Derivative algebra (abstract algebra)|derivative operator]], <sup>D</sup>. Hence we can form a [[Abstract algebra|derivative algebra]] '''''D'''''('''''A''''') with the same underlying Boolean algebra as '''''A''''' by using the closure operator as a derivative operator.

Thus interior algebras are [[Derivative algebra (abstract algebra)|derivative algebras]]. From this perspective, they are precisely the [[variety (universal algebra)|variety]] of derivative algebras satisfying the identity ''x''<sup>D</sup> ≥ ''x''. Derivative algebras provide the appropriate [[Lindenbaum–Tarski algebra|algebraic semantics]] for the modal logic '''wK4'''. Hence derivative algebras stand to topological [[derived set (mathematics)|derived set]]s and '''wK4''' as interior/closure algebras stand to topological interiors/closures and '''S4'''.
 
Given a derivative algebra '''''V''''' with derivative operator <sup>D</sup>, we can form an interior algebra {{math|1='''''I'''''('''''V''''')}} with the same underlying Boolean algebra as '''''V''''', with interior and closure operators defined by {{math|1=''x''<sup>I</sup> = ''x''·''x'' ′ <sup>D</sup> ′}} and {{math|1=''x''<sup>C</sup> = ''x'' + ''x''<sup>D</sup>}}, respectively. Thus every derivative algebra can be regarded as an interior algebra. Moreover, given an interior algebra '''''A''''', we have {{math|1='''''I'''''('''''D'''''('''''A''''')) = '''''A'''''}}. However, {{math|1='''''D'''''('''''I'''''('''''V''''')) = '''''V'''''}} does ''not'' necessarily hold for every derivative algebra '''''V'''''.