Product Code Database
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 algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of .
where
is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities:
xI is called the interior of x.
The dual of the interior operator is the closure operator C defined by xC = (( x′)I)′. xC is called the closure of x. By the principle of duality, the closure operator satisfies the identities:
If the closure operator is taken as primitive, the interior operator can be defined as xI = (( x′)C)′. 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, C⟩, where ⟨ S, ·, +, ′, 0, 1⟩ is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras form 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 following the work of Wim Blok.
An interior algebra is called Boolean if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of trivial interior algebras, which are the single element interior algebras characterized by the identity 0 = 1.
and extend it to an interior algebra
where I is the usual topological interior operator. For all S ⊆ X it is defined by
For all S ⊆ X the corresponding closure operator is given by
SI is the largest open subset of S and SC 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 isomorphism to an interior algebra of the form A( X) for some topological space X. Moreover, every interior algebra can be embedding 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
we can define a complete topomorphism
by
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 theory of complete atomic interior algebras and complete topomorphisms then Top and Cit are dually isomorphic and is a functor that is a dual isomorphism of categories. A( f) is a homomorphism if and only if f is a continuous open map.
Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties:
where ⟨ B, ·, +, ′, 0, 1⟩ is a Boolean algebra as usual, and T is a unary relation on B (subset of B) such that:
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
we can define an interior operator on B by 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 relations, so that standard results from universal algebra apply.
A neighbourhood function on a Boolean algebra is a mapping N from its underlying set B to its set of filters, such that:
The mapping N of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function N on a Boolean algebra with underlying set B, we can define an interior operator by thereby obtaining an interior algebra. will then be precisely the filter of neighbourhoods of x in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.
In terms of neighbourhood functions, the open elements are precisely those elements x such that . In terms of open elements if and only if there is an open element z such that .
Neighbourhood functions may be defined more generally on semilattice producing the structures known as neighbourhood (semi)lattices. Interior algebras may thus be viewed as precisely the Boolean neighbourhood lattices i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.
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 logic □ ( necessarily), while the closure operator corresponds to ◊ ( possibly). This construction is a special case of a more general result for 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 correspond to those that are only false if they are necessarily false.
Because of their relation to S4, interior algebras are sometimes called S4 algebras or Lewis algebras, after the logician C. I. Lewis, who first proposed the modal logics S4 and S5.
Given a Preorder X = ⟨ X, «⟩ we can construct an interior algebra
from the power set Boolean algebra of X where the interior operator I is given by
The corresponding closure operator is given by
SI is the set of all worlds inaccessible from worlds outside S, and SC is the set of all worlds accessible from some world in S. Every interior algebra can be embedding in an interior algebra of the form B( X) for some Preorder 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 and Kripke frames. In this regard, interior algebras are particularly interesting because of their connection to topology. The construction provides the Preorder X with a topology, the Alexandrov topology, producing a topological space T( X) whose open sets are:
The corresponding closed sets are:
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)).
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 Kripke 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 Kripke semantics using monadic universal and existential quantification, respectively, without reference to an accessibility relation.
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 relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and S4, in which one can interpret theories of intuitionistic logic as S4 theories closed under logical truth. The one-to-one correspondence between Heyting algebras and interior algebras generated by their open elements reflects the correspondence between extensions of intuitionistic logic and normal extensions of the modal logic S4.Grz.
Thus interior algebras are derivative algebras. From this perspective, they are precisely the variety of derivative algebras satisfying the identity xD ≥ x. Derivative algebras provide the appropriate algebraic semantics for the modal logic wK4. Hence derivative algebras stand to topological derived sets and wK4 as interior/closure algebras stand to topological interiors/closures and S4.
Given a derivative algebra V with derivative operator D, we can form an interior algebra with the same underlying Boolean algebra as V, with interior and closure operators defined by and , respectively. Thus every derivative algebra can be regarded as an interior algebra. Moreover, given an interior algebra A, we have . However, does not necessarily hold for every derivative algebra V.
Whereas the Jónsson–Tarski generalization of Stone duality applies to Boolean algebras with operators in general, the connection between interior algebras and topology allows for another method of generalizing Stone duality that is unique to interior algebras. An intermediate step in the development of Stone duality is Stone's representation theorem, which represents a Boolean algebra as a field of sets. The Stone topology of the corresponding Boolean space is then generated using the field of sets as a topological basis. Building on the topological semantics introduced by Tang Tsao-Chen for Lewis's modal logic, McKinsey and Tarski showed that by generating a topology equivalent to using only the complexes that correspond to open elements as a basis, a representation of an interior algebra is obtained as a topological field of sets—a field of sets on a topological space that is closed with respect to taking interiors or closures. By equipping topological fields of sets with appropriate morphisms known as field maps, C. Naturman showed that this approach can be formalized as a category theoretic Stone duality in which the usual Stone duality for Boolean algebras corresponds to the case of interior algebras having redundant interior operator (Boolean interior algebras).
The pre-order obtained in the Jónsson–Tarski approach corresponds to the accessibility relation in the Kripke semantics for an S4 theory, while the intermediate field of sets corresponds to a representation of the Lindenbaum–Tarski algebra for the theory using the sets of possible worlds in the Kripke semantics in which sentences of the theory hold. Moving from the field of sets to a Boolean space somewhat obfuscates this connection. By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and Kripke frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics. The pre-order can be obtained as the specialization pre-order of the McKinsey–Tarski topology. The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates. Via a functor that instead replaces the pre-order with its corresponding Alexandrov topology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico-reflection of the McKinsey–Tarski topology. The approach of formulating a topological duality for interior algebras using both the Stone topology of the Jónsson–Tarski approach and the Alexandrov topology of the pre-order to form a bi-topological space has been investigated by G. Bezhanishvili, R.Mines, and P.J. Morandi. The McKinsey–Tarski topology of an interior algebra is the intersection of the former two topologies.
|
|
