In abstract algebra, an **interior algebra** is an algebraic structure of the signature - <
*A*, ·, +, ', 0, 1, ^{I}> where - <
*A*, ·, +, ', 0, 1> is a Boolean algebra and ^{I} is a unary operator, the **interior operator**, satisfying the identities: *x*^{I} ≤ *x* *x*^{II} = *x*^{I} - (
*xy*)^{I} = *x*^{I}*y*^{I} - 1
^{I} = 1 *x*^{I} is called the **interior** of *x*. Interior algebras play the same role for the modal logic *S4* that Boolean algebras play for ordinary propositional logic and can be regarded as a variety of modal algebras. They also play the same role for topology that Boolean algebras play for set theory. The dual of the interior operator is the **closure operator** ^{C} defined by *x*^{C} = *x* ' ^{I} '. By the principle of duality, the closure operator satisfies the identities: *x*^{C} ≥ *x* *x*^{CC} = *x*^{C} - (
*x* + *y*)^{C} = *x*^{C} + *y*^{C} - 0
^{C} = 0 *x*^{C} is called the **closure** of *x*. The interior operator is recoverable from the closure operator via the identity *x*^{I} = *x* ' ^{C} '. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator. In this formulation one considers algebraic structures of the form - <
*A*, ·, +, ', 0, 1, ^{C}> called **closure algebras** where - <
*A*, ·, +, ', 0, 1> is a Boolean algebra and ^{C} satisfies the properties of a closure operator listed above. By the principle of duality, closure algebras are entirely equivalent to interior algebras. (The closure operator formulation was used in the early literature on the subject, but the interior operator formulation became the standard in later literature.) ## Open and closed elements
Elements of an interior algebra satisfying the condition *x*^{I} = *x* are called **open**. The complements of open elements are called **closed** and are characterized by the condition *x*^{C} = *x*. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called **regular open** and closures of open elements are called **regular closed**. Elements which are both open and closed are called **clopen**. 0 and 1 are clopen.
## Morphisms of Interior Algebras ### Homomorphisms Since interior algebras are algebraic structures we can speak of interior algebra homomorphisms. Given two interior algebras **A** and **B**, a map *f* : **A** → **B** is an interior algebra homomorphism if and only if it is a homomorphism between the underlying Boolean algebras of **A** and **B** and in addition preserves interiors (and hence equivalently, preserves closures) i.e. *f*(*x*^{I}) = *f*(*x*)^{I} *f*(*x*^{C}) = *f*(*x*)^{C} ### Topomorphisms Another important, and more general, class of morphisms between interior algebras are the **topomorphisms**. A map *f* : **A** → **B** is a **topomorphism** if and only if it is a homomorphism between the underlying Boolean algebras of **A** and **B** and in addition preserves open elements (and hence equivalently, preserves closed elements) i.e. - 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**. Every interior algebra homomorphism is a topomorphism but not every topomorphism is an interior algebra homomorphism.
## Relationships to other areas of mathematics Given a topological space **X** = <*X*, *T*> one can form the power set Boolean algebra of *X* - <
*P*(*X*), ∩, ∪, ', ø, *X*> and extend it to an interior algebra **A**(**X**) = <*P*(*X*), ∩, ∪, ', ø, *X*, ^{I}> where ^{I} is the usual topological interior operator defined by *S* ^{I} = { *O* ∈ *T* : *O* ⊆ *S* } for all *S* ⊆ *X* The corresponding closure operator is given by *S* ^{C} = { *C* : *S* ⊆ *C* and *C* is closed in **X** } for all *S* ⊆ *X* *S* ^{I} is the largest open subset of *S* and *S* ^{C} is the smallest closed superset of *S* in **X**. The open, closed, regular open, regular closed and clopen elements of the interior algebra **A**(**X**) are just the open, closed, regular open, regular closed and clopen subsets of **X** respectively in the usual topological sense. Every complete atomic interior algebra is isomorphic to an interior algebra of the form **A**(**X**) for some topological space **X**. Moreover every interior algebra can be embedded in such an interior algebra giving a representation of an interior algebra as a **topological field of sets**. The properties of the structure **A**(**X**) are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called **topo-Boolean algebras** or **topological Boolean algebras**. Given a continuous map between two topological spaces *f* : **X** → **Y** we can define a complete topomorphism -
**A**(*f*) : **A**(**Y**) → **A**(**X**) by **A**(*f*)(*S*) = *f* ^{-1}[*S*] for all subsets *S* of **Y**. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If **Top** is the category of topological spaces and continuous maps and **Cit** is the category of complete atomic interior algebras and complete topomorphisms then **Top** and **Cit** are dually isomorphic and **A** : **Top** → **Cit** is a contravariant functor that is a dual isomorphism of categories. **A**(*f*) is a homomorphism if and only if *f* is a continuous open map.
#### Generalized topology The modern formulation of topological spaces in terms of topologies of open subsets, motivates an alternative formulation of interior algebras: A **generalized topological space** is an algebraic structure of the form - <
*B*, ·, +, ', 0, 1, *T*> where - <
*B*, ·, +, ', 0, 1> is a Boolean algebra and *T* is a unary relation on *B* (subset of *B*) such that - 0,1 ∈
*T* *T* is closed under arbitrary joins *T* is closed under finite meets - For every element
*b* of *B*, the join ∑{*a* ∈*T* : *a* ≤ *b*} exists *T* is said to be a **generalized topology** in the Boolean algebra. Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space - <
*B*, ·, +, ', 0, 1, *T*> we can define an interior operator on *B* by *b* ^{I} = ∑{*a* ∈*T* : *a* ≤ *b*} thereby producing an interior algebra whose open elements are precisely *T*. Thus generalized topological spaces are equivalent to interior algebras. Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added unary relation and standard results from universal algebra apply to them.
#### Neighbourhood mappings and neighbourhood lattices Given a theory (set of formal sentences) *M* in the modal logic *S4*, we can form its Lindenbaum-Tarski algebra **L**(*M*) = <*M* / ~, ∧, ∨, ¬, *F*, *T*, □> where ~ is the equivalence relation on sentences in *M* given by *p* ~ *q* if and only if *p* and *q* are logically equivalent in *M*, and *M* / ~ is the set of equivalence classes under this relation. Then **L**(*M*) is an interior algebra. The interior operator in this case corresponds to the modal operator □ (**necessarily**) while the closure operator corresponds to ◊ (**possibly**). This construction is a special case of a more general result for modal algebras and modal logic. The open elements of **L**(*M*) correspond to sentences that are only true if they are **necessarily** true while the closed elements corresond to those that are only false if they are **necessarily** false. Because of their relationship to the modal logic *S4*, interior algebras have also been called **S4 algebras** or **Lewis algebras** (after the logician C. I. Lewis who classified the modal logics *S1* - *S5*).
Since interior algebras are (normal) Boolean algebras with operators they can be represented by fields of sets on appropriate relational structures. In particular since they are modal algebras they can be represented as fields of sets on a set with a single binary relation, called a modal frame in this context. The modal frames corresponding to interior algebras are precisely the preordered sets. Preordered sets (also called *S4-frames*) provide the possible world semantics of the modal logic *S4* and the connection between interior algebras and preorders is deeply related to their connection with modal logic. Given a preordered set **X** = <*X*, « > we can construct an interior algebra **B**(**X**) = <*P*(*X*), ∩, ∪, ', ø, *X*, ^{I}> from the power set Boolean algebra of *X* where the interior operator ^{I} is given by *S* ^{I} = { *x* ∈ *X* : for all *y* ∈ *X*, *x* « *y* implies *y* ∈ *S* } for all *S* ⊆ *X*. The corresponding closure operator is given by *S* ^{C} = { *x* ∈ *X* : there exists a *y* ∈ *S* with *x* « *y* } for all *S* ⊆ *X*. *S* ^{I} is the set of all *worlds* inaccessible from *worlds* outside *S*, and *S* ^{C} is the set of all *worlds* accessible from some *world* in *S*. Every interior algebra can be embedded in an interior algebra of the form **B**(**X**) for some preordered set **X** giving the above mentioned representation as a field of sets (a **preorder field**). This construction and representation theorem is a special case of the more general result for modal algebras and modal frames. The case for interior algebras is particularly interesting because of their connection to topology. The construction provides the preordered set **X** with a topology (the Alexandrov topology) producing a topological space **T**(**X**) with open sets given by - {
*O* ⊆ *X* : for all *x* ∈ *O* and all *y* ∈ *X*, *x* « *y* implies *y* ∈ *O* } and corresponding closed sets given by - {
*C* ⊆ *X* : for all *x* ∈ *C* and all *y* ∈ *X*, *y* « *x* implies *y* ∈ *C* }. In other words the open sets are the ones whose *worlds* are inaccessible from outside (the **up-sets**) and the closed sets are the ones for which every outside *world* is inaccessible from inside (the **down-sets**). Moreover **B**(**X**) = **A**(**T**(**X**)).
Any Monadic Boolean algebra can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The Monadic Boolean algebras are then precisely the variety of interior algebras satisfying the identity *x*^{IC} = *x*^{I}. In other words they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the semisimple interior algebras. They are also the interior algebras corresponding to the modal logic *S5* and in this context have also been called **S5 algebras**. In the relationship between preordered sets and interior algebras they correspond to the case where the preorder is an equivalence relation, reflecting the fact that such preordered sets provide the possible world semantics for *S5*. This also reflects the relationship between the monadic logic of quantification (for which monadic Boolean algebras provide an algebraic description) and *S5* where the modal operators □ (**necessarily**) and ◊ (**possibly**) can be interpreted in the possible world semantics using monadic universal and existential quantification respectively.
### Heyting algebras The open elements of an interior algebra form a Heyting algebra and the closed elements form a dual Heyting algebra. The regular open elements and regular closed elements correspond to the pseudo-complemented elements and dual pseudo-complemented elements of these algebras respectively and thus form Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every Heyting algebra can be represented as the open elements of an interior algebra. Heyting algebras play the same role for intuitionistic logic that interior algebras play for the modal logic *S4* and Boolean algebras play for propositional logic. The relationship between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and the modal logic *S4* in which one can interpret theories in intuitionistic logic as theories in *S4* closed under necessity.
### Derivative algebras Given an interior algebra **A**, the closure operator obeys the axioms of a derivative operator and so we can form a derivative algebra **D**(**A**) with the same underlying Boolean algebra as **A**, by defining a derivative operator by *x*^{D} = *x*^{C}. We can thus regard interior algebras as being derivative algebras. In this perspective they are precisely the variety of derivative algebras satisfying the identity *x*^{D} ≥ *x*. Given a derivative algebra **V** with derivative operator ^{D}, we can form an interior algebra **I**(**V**) with the same underlying Boolean algebra as **V** and interior and closure operators defined by *x*^{I} = *x*·*x* ' ^{D} ' and *x*^{C} = *x* + *x*^{D} respectively. Thus every derivative algebra can be regarded as an interior algebra. Moreover given an interior algebra **A** we have **I**(**D**(**A**)) = **A**. However we do **not** necessarily have **D**(**I**(**V**)) = **V** for every derivative algebra **V**. The relationship between interior algebras and derivative algebras reflects the relationship between the modal logic *S4* and the modal logic *wK4* for which derivative algebras provide the appropriate algebraic semantics. It also reflects the relationship in topology between interiors and closures and derived sets.
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*Intuitionistic logic and modality via topology*, Annals of Pure and Applied Logic, 127, 155-170, 2004 - McKinsey, J.C.C. and Tarski, A.,
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