Editing Interior algebra (section)

== Morphisms of interior algebras ==

=== Homomorphisms ===
Interior algebras, by virtue of being [[algebraic structure]]s, have [[homomorphism]]s. Given two interior algebras ''A'' and ''B'', a map ''f'' : ''A'' → ''B'' is an '''interior algebra homomorphism''' [[if and only if]] ''f'' is a homomorphism between the underlying Boolean algebras of ''A'' and ''B'', that also preserves interiors and closures. Hence:
*''f''(''x''<sup>I</sup>) = ''f''(''x'')<sup>I</sup>;
*''f''(''x''<sup>C</sup>) = ''f''(''x'')<sup>C</sup>.

=== Topomorphisms ===
Topomorphisms are another important, and more general, class of [[morphism]]s between interior algebras. A map ''f'' : ''A'' → ''B'' is a topomorphism if and only if ''f'' is a homomorphism between the Boolean algebras underlying ''A'' and ''B'', that also preserves the open and closed elements of ''A''. Hence:
* If ''x'' is open in ''A'', then ''f''(''x'') is open in ''B'';
* If ''x'' is closed in ''A'', then ''f''(''x'') is closed in ''B''.
(Such morphisms have also been called ''stable homomorphisms'' and ''closure algebra semi-homomorphisms''.) Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism.

=== Boolean homomorphisms ===
Early research often considered mappings between interior algebras that were homomorphisms of the underlying Boolean algebras but that did not necessarily preserve the interior or closure operator. Such mappings were called '''Boolean homomorphisms'''. (The terms ''closure homomorphism'' or ''topological homomorphism'' were used in the case where these were preserved, but this terminology is now redundant as the standard definition of a homomorphism in [[universal algebra]] requires that it preserves all operations.) Applications involving countably complete interior algebras (in which countable [[meet (order theory)|meet]]s and [[join (order theory)|join]]s always exist, also called ''σ-complete'') typically made use of countably complete Boolean homomorphisms also called '''Boolean σ-homomorphisms'''—these preserve countable meets and joins.

=== Continuous morphisms ===
The earliest generalization of continuity to interior algebras was [[Roman Sikorski|Sikorski]]'s, based on the [[inverse image]] map of a [[continuous map]]. This is a Boolean homomorphism, preserves unions of sequences and includes the closure of an inverse image in the inverse image of the closure. Sikorski thus defined a ''continuous homomorphism'' as a Boolean σ-homomorphism ''f'' between two σ-complete interior algebras such that ''f''(''x'')<sup>C</sup> ≤ ''f''(''x''<sup>C</sup>). This definition had several difficulties: The construction acts [[Functor#Covariance and contravariance|contravariantly]] producing a dual of a continuous map rather than a generalization. On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ-complete homomorphisms but included σ-completeness in his axioms for ''closure algebras''.) Later J. Schmid defined a '''continuous homomorphism''' or '''continuous morphism''' for interior algebras as a Boolean homomorphism ''f'' between two interior algebras satisfying ''f''(''x''<sup>C</sup>) ≤ ''f''(''x'')<sup>C</sup>. This generalizes the forward image map of a continuous map—the image of a closure is contained in the closure of the image. This construction is [[Functor#Covariance and contravariance|covariant]] but not suitable for [[category theoretic]] applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C. Naturman returned to Sikorski's approach while dropping σ-completeness to produce topomorphisms as defined above. In this terminology, Sikorski's original "continuous homomorphisms" are σ-complete topomorphisms between σ-complete interior algebras.)