Editing Interior algebra (section)

==Definition==
An '''interior algebra''' is an [[algebraic structure]] with the [[signature (logic)|signature]]

:⟨''S'', ·, +, ′, 0, 1, <sup>I</sup>⟩

where

:⟨''S'', ·, +, ′, 0, 1⟩

is a [[Boolean algebra (structure)|Boolean algebra]] and postfix <sup>I</sup> designates a [[unary operator]], the '''interior operator''', satisfying the identities:

# ''x''<sup>I</sup> ≤ ''x''
# ''x''<sup>II</sup> = ''x''<sup>I</sup>
# (''xy'')<sup>I</sup> = ''x''<sup>I</sup>''y''<sup>I</sup>
# 1<sup>I</sup> = 1

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

The [[duality (order theory)|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 [[duality (order theory)|principle of duality]], the closure operator satisfies the identities:

# ''x''<sup>C</sup> ≥ ''x''
# ''x''<sup>CC</sup> = ''x''<sup>C</sup>
# (''x'' + ''y'')<sup>C</sup> = ''x''<sup>C</sup> + ''y''<sup>C</sup>
# 0<sup>C</sup> = 0

If the closure operator is taken as primitive, the interior operator can be defined as ''x''<sup>I</sup> = ((''x''′)<sup>C</sup>)′. 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, <sup>C</sup>⟩, where ⟨''S'', ·, +, ′, 0, 1⟩ is again a Boolean algebra and <sup>C</sup> satisfies the above identities for the closure operator. Closure and interior algebras form [[duality (order theory)|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{{fact|date=February 2023}}<!--By the way, the most recent works in the reference list below use "closure algebras". The same for the book I have consulted,  P. Blackburn, M. de Rijke & Y. Venema Modal Logic --> following the work of [[Wim Blok]].