A Hierarchical Lattice Closure Approach toAbstract Rough Approximation Spaces�

Gianpiero Cattaneo and Davide Ciucci

Dipartimento di Informatica, Sistemistica e ComunicazioneUniversita di Milano – Bicocca, Viale Sarca 336–U14, I–20126 Milano, Italia

{cattang,ciucci}@disco.unimib.it

Abstract. A hierarchy of closure operators on the abstract context oflattice structures is investigated, and compared to the abstract approachto rough approximation spaces. In particular, the Tarski, the Kuratowskiand the Halmos closures are treated, with the corresponding models ofcovering, topological and partition spaces.

1 Closure Operation in Lattice and Rough ApproximationSpaces

In [3,4] a first investigation about closure operations in a lattice context has beendone, based on the proof of the equivalence of (weak) closure with the notion ofrough approximation space according to the definition given in [2]. In this paperwe give a further in depth analysis of this argument. First of all, let us recall thenotion of closure, developed by Tarski in his study in consequences in logic [16].

Definition 1. A Tarski closure lattice is a structure 〈Σ, ∧, ∨, ′, ∗, 0, 1〉 where:

(Cl-dM1) The substructure 〈Σ, ∧, ∨, ′, 0, 1〉 is a de Morgan lattice, i.e., a boundedlattice, equipped with a de Morgan complementation mapping ′ : Σ �→Σ that satisfies, for arbitrary a, b ∈ Σ, the conditions: (dM1) a = a′′;(dM2) (a ∨ b)′ = a′ ∧ b′.

(Cl-dM2) The mapping ∗ : Σ → Σ is a Tarski closure operation, that is, itsatisfies:

(C1) 0∗ = 0 (normalized)(C2) a ≤ a∗ (increasing)(C3) a∗ = a∗∗ (idempotent)(C4) a∗ ∨ b∗ ≤ (a ∨ b)∗ (sub–additive)

The subset of closed elements is defined as the collection of elements whichare equal to their closure: formally, C(Σ) = {a ∈ Σ : a = a∗} . This set is not� The author’s work has been supported by MIUR\PRIN project ”Automata and

Formal languages: mathematical and application driven studies”.For lack of space, this is an extended abstract of a forthcoming larger version.

G. Wang et al. (Eds.): RSKT 2008, LNAI 5009, pp. 363–370, 2008.c© Springer-Verlag Berlin Heidelberg 2008

364 G. Cattaneo and D. Ciucci

empty since 0, 1 are closed elements. Condition (C3) says that for any elementa ∈ Σ the corresponding a∗ is closed; this element is called the closure of a.With respect to the notion of closure in a de Morgan lattice, the dual notion isthe following one.

Theorem 1. Suppose aTarski closure (deMorgan) lattice T =〈Σ, ∧, ∨, ′, ∗, 0, 1〉.The mapping o : Σ → Σ defined by the law

∀a ∈ Σ, ao := ((a′)∗)′ (1)

is a (Tarski) interior operation, i.e., it satisfies the followings:

(I1) 1o = 1 (normalized)(I2) ao ≤ a (decreasing)(I3) ao = aoo (idempotent)(I4) (a ∧ b)o ≤ ao ∧ bo (sub–multiplicative)

Analogously to the closure case, the structure T i = 〈Σ, ∧, ∨, ′, o, 0, 1〉 is said tobe a Tarski interior lattice.

The subset of open elements is defined as the collection of elements which areequal to their interior. Formally, O(Σ) = {a ∈ Σ : a = ao} . Also this set isnot empty since the two elements 0, 1 are open too. Condition (I3) says that forevery a ∈ Σ the element ao, called the interior of a, is open.

Other important concepts, which is possible to introduce are: the set of clopenelements: CO(Σ) := C(Σ) ∩ O(Σ), which is not empty and, in general, do notcoincide with the set of open or closed elements; the so–called exterior of anelement a ∈ Σ defined as the open element e(a) := (a∗)′ ∈ O(Σ); the boundarydefined as the closed element b(a) := (ao)′ ∧ a∗ ∈ C(Σ).

In order to grasp the intuitive aspects of the abstract notion of rough approx-imation space, as introduced in [2], we have to recall the following result.

Theorem 2. Suppose a (de Morgan) lattice equipped with a Tarski interiorand a Tarski closure operations A = 〈Σ, o, ∗ 〉. Then, the structure A� :=〈Σ, O(Σ), C(Σ)〉 is a (de Morgan) rough approximation space in the sense that,defined for any arbitrary approximable element a ∈ Σ its lower approximationas l(a) := ao (the interior of a) and its upper approximation as u(a) := a∗ (theclosure of a), the following hold:

(In1) l(a) ∈ O(Σ) (In2) l(a) ≤ a (In3) ∀β ∈ O(Σ), β ≤ a ⇒ β ≤ l(a);(Up1) u(a) ∈ C(Σ) (Up2) a ≤ u(a) (Up3) ∀γ ∈ C(Σ), a ≤ γ ⇒ u(a) ≤ γ.

Any structure 〈Σ, C(Σ), u 〉 (resp., 〈Σ, O(Σ), l 〉 satisfying the conditions(Up1)–(Up3) (resp., (In1)–(In3)) is called an upper (resp., a lower) rough ap-proximation space, and the mapping l : Σ �→ O(Σ) (resp., u : Σ �→ C(Σ)) the

A Hierarchical Lattice Closure Approach 365

lower (resp., upper) approximation map. Relatively to these last structures, theelement of C(Σ) (resp., O(Σ)) are also called upper (resp., lower) crisp elements.

In general, the above outlined structures of Tarski closure and induced interiorlattices allow one to introduce the important notion of rough approximationmapping r : Σ �→ C(Σ)× O(Σ) which associates to any element a ∈ Σ its open–closed rough approximation r(a) = (ao, a∗), with ao ∈ O(Σ), a∗ ∈ C(Σ), anda0 ≤ a ≤ a∗ (see I2 and C2). In this context, an element e is said to be crisp (orexact, also sharp) iff r(e) = (e, e), and this is equivalent to ask that e ∈ CO(Σ).

Some remarks.

(i) The non–equational conditions of Theorem 2 capture the intuitive aspectsof an expected rough approximation space: the lower (resp., upper) approx-imations l(a) (resp., u(a)) is the best approximation of the approximableelement a from the bottom (resp., top) by open (resp. closed) elements.

(ii) Theorem 2 is only a part of a more complete version discussed in [3] whichcan be summarized by the statement that the concrete category of deMorgan lattice with Tarski closure (and induced interior) operator and theone of de Morgan lattice with upper (and induced inner) approximationmap are categorical equivalent (isomorphic) between them.

(iii) Thus, if one works in the context of a generalization of the closure operatorconditions, for instance weakening (or suppressing) some of the above con-ditions (C1)–(C4) of definition 1, then some of the “intuitive” conditionsexpressed by (Up1)–(Up3) are correspondingly lost, or modified.

2 A Hierarchy of Closure Operators

Starting from the original Tarski closure given in definition 1, it is possible tointroduce a hierarchy of closure operators with increasingly strong behaviour.

Definition 2. A closure operation on a de Morgan lattice is said to be topo-logical iff the sub–additive condition (C4) is substituted by the stronger additiveproperty: (C4T) a∗ ∨ b∗ = (a ∨ b)∗.

Citing from Rasiowa–Sikorski [14]: “The closure operation ∗ satisfies the con-ditions (C1)–(C3) and (C4T). These Axioms are due to Kuratowski [7]. For adetailed exposition of the theory of topological spaces see e.g. Kelley [8], Ku-ratowski [9,10].” In agreement with this quotation, in the sequel we shall callKuratowski closure operation any topological closure on a de Morgan lattice.A de Morgan lattice equipped with a Kuratowski closure operations is called aKuratowski closure lattice.

The Kuratowski interior operator dually defined according to the equation(1) satisfies the multiplicative property: (I4T) ao ∧ bo = (a ∧ b)o.

Definition 3. A Kuratowski closure operation on a de Morgan lattice is a Hal-mos closure iff the idempotent condition (C3) is substituted by the stronger one:(sC3) a∗ ′ ∗ = a∗ ′. A Halmos closure lattice is any de Morgan lattice equippedwith a Halmos closure operation.

366 G. Cattaneo and D. Ciucci

This definition has been introduced by Halmos in [5] (collected in [6]) on the baseof a Boolean algebra structure as the algebraic formalization of the existentialquantifier. In the present context, we generalize this structure to the case ofde Morgan lattices. The following result assures that the Halmos closure is aKuratowski closure too.

Proposition 1. Let Σ be a de Morgan lattice with an operation ∗ : Σ �→ Σwhich satisfies conditions (C1), (C2), and (C4T). Then the following implicationholds: (sC3) ∀a ∈ Σ, a∗ ′ ∗ = a∗ ′ implies (C3) ∀a ∈ Σ, a∗ = a∗∗.

The following property characterizes de Morgan lattices with Halmos closure.

Proposition 2. Let Σ be a de Morgan lattice with a topological (i.e., Kura-towski) closure operation. Then the following are equivalent:

1. the topological closure operation satisfies condition (sC3), i.e., it is Halmos;2. the collection of closed elements and the collection of open elements coincide:

C(Σ) = O(Σ).

3 Tarski, Kuratowski and Halmos Closure Models on aConcrete Universe

In this section we investigate three concrete examples of the three abstract no-tions of closure, all based on a concrete nonempty set, the universe of the dis-course X , and its power set P(X), the Boolean lattice of all subsets of X .

3.1 Coverings of a Universe as Models of Tarski Closure

First of all let us consider a covering of the universe X , i.e., a family γ = {Ci ∈P(X) : i ∈ I} of nonempty subsets of X (indexed by the index set I) whichsatisfies the covering condition: X = ∪{Ci ∈ γ : i ∈ I} .

Open sets are defined as the set theoretic union of subsets from the coveringplus the empty set ∅. Denoting by Oγ(X) their collection, we have that

L ∈ Oγ(X) iff ∃{Cj ∈ γ : j ∈ I} : L =⋃

j∈I

Cj (2)

The following properties of an open family hold:

(PO1) Oγ(X) contains both the empty set and the whole universe, and(PO2) it is closed with respect to arbitrary set theoretic union.

To any subset A of the universe X it is possible to assign the open set lγ(A) =∪{L ∈ Oγ(X) : L ⊆ A}. The main result is the following.

Proposition 3. The mapping lγ : P(X) �→ P(X), A → lγ(A) is a Tarskiinterior operator, or from an equivalent point, it is a lower approximation map.

A Hierarchical Lattice Closure Approach 367

The lower rough approximation space with respect to the covering γ of X isthen LRγ = 〈P(X), Oγ(X), lγ〉, where the lower (open) approximation of A isthe interior Ao := lγ(A). In particular, from the point of view of a rough ap-proximation space: the role of the bounded de Morgan lattice Σ of approximableelements is played by the Boolean lattice P(X); the set of open elements Oγ(X)is the lattice of lower crisp elements; for any subset A of X (the approximableelement) the corresponding lower approximation is its interior, i.e., the greatestopen set contained in A.

The dual of the covering γ is the anti–covering γd := {D ∈ P(X) : ∃C ∈γ s.t D = Cc}, which satisfies the disjointness condition: ∅ = ∩{D : D ∈ γd} .A closed subset is then the set theoretic intersection of elements from the dualcovering. The collection of all closed sets induced by the covering γ will bedenoted by Cγ(X) and so

U ∈ Cγ(X) iff ∃{Ds ∈ γd : s ∈ S} : U =⋂

s∈S

Ds (3)

with U ∈ Cγ(X) iff ∃L ∈ Oγ(X) s.t. U = Lc . Of course, the following prop-erties of the closed family holds:

(PC1) Cγ(X) contains both the empty set and the whole universe, and(PC2) it is closed with respect to arbitrary set theoretic intersection.

These properties qualify Cγ(X) as a Moore family of subsets of X (see [1, p. 111]).The upper (closed) approximation of any subset A of X is uγ(A) = ∩{U ∈Cγ(X) : A ⊆ U} and we have the following result.

Proposition 4. The mapping uγ : P(X) �→ P(X), A → uγ(A) is a Tarskiclosure operator or, equivalently, it is an upper approximation map.

In this closure framework the corresponding upper approximation space withrespect to the covering γ is then CRγ = 〈P(X), Cγ(X), uγ〉, where P(X) is thecollection of approximable elements; the collection Cγ(X) of all closed sets is thelattice of upper crisp elements; for any A ⊆ X (the approximable element) thecorresponding upper approximation is its closure, i.e., the smallest closed setcontaining A usually denoted by A∗ := u(A).

The rough approximation space induced by the covering γ is then the structureRT = 〈P(X), Oγ(X), Cγ(X), lγ , uγ〉 where the rough approximation mappingrγ : P(X) �→ Oγ(X)×Cγ(X) assigns to any subset A of the universe X its roughapproximation as the open–closed pair rγ(A) = (Ao, A∗), with Ao ⊆ A ⊆ A∗.Note that crisp elements are just the clopen CγOγ(Σ) = Cγ(Σ) ∩ Oγ(Σ).

Borrowing some definitions from the (discussed in the sequel) partition case,in the present covering context it is also possible to introduce the two openapproximations of any subset A as

l(op)γ (A) := ∪{C ∈ γ : C ⊆ A} = lγ(A) (4a)

u(op)γ (A) := ∪{D ∈ γ : D ∩ A �= ∅} �= uγ(A) (4b)

368 G. Cattaneo and D. Ciucci

This second definition, in general does not give the same result, since u(op)γ (A) ∈

O(X) is an open set (as set theoretic union of open sets), whereas uγ(A) =A∗ ∈ C(X) is a closed set (as set theoretic intersection of closed sets). From aterminological point of view, uγ(A) can be called the closed upper approximationand u

(op)γ (A) the open upper approximation of A.

Proposition 5. In any covering space the following inequality between upperapproximations holds: ∀A ∈ P(X), uγ(A) ⊆ u

(op)γ (A) .

3.2 Topologies of a Universe as Models of Kuratowski Closure

Let us recall that a topological space can be introduced as a pair τ = (X, B)consisting of a set X and a base for a topology (simply open base), i.e., a collec-tion B = {Bi ∈ P(X) : i ∈ I} of subsets of X , each of which is an open granule(or adopting the topological terminology, see [15, p. 99], a basic open set), suchthat the following hold:

(C) covering condition: X = ∪{Bi ∈ B : i ∈ I} ;(OB) open base condition: for any pair Bi, Bj of subsets of the base B for which

Bi ∩ Bj �= ∅ a collection {Bih∈ B : h ∈ H} of elements from the base

exists such that Bi ∩ Bj = ∪{Bih∈ B : h ∈ H}.

This constitutes a strengthening of the notion of covering owing to the furthercondition (OB). Thus, similarly to the covering case, see equation (2), we intro-duce the notion of topological open sets as any subset of X which is either a settheoretic union of elements from the open base B or the empty set ∅. In this waywe have induced a topological space defined as a pair (X, O(X)) consisting of anonempty set X equipped with a family of open subsets O(X)In particular anyelement of the open basis is an open set, that is B ⊆ O(X).

Further, we can construct the dual structure τd = (X, K), where K := {K ∈P(X) : ∃B ∈ B s.t. K = Bc} is the dual base of closed granules. This collectionK is a base of closed sets (simply, closed base, see [15, p. 112]) for a topology onX in the sense that the following hold:

(DB) disjointness condition: ∅ = ∩{K ∈ P(X) : K ∈ K};(CB) closed base condition: for any pair Ki, Kj of subsets of the closed base K

for which Ki ∪ Kj �= X a collection {Kiv ∈ K : v ∈ V } of elements fromthe closed base exists such that Ki ∪ Kj = ∩{Kiv ∈ K : v ∈ V }.

Similarly to the covering case, see equation (3), a topological closed set is eitherthe set theoretic intersection of elements from the closed base or the whole spaceX . Trivially, a subset of X is closed iff it is the set theoretic complement of anopen set: C ∈ C(X) iff ∃O ∈ O(X) : C = Oc . All the elements of the closedbase are also closed, that is K ⊆ C(X).

Note that, as a particular case of closure, also in the now treated case of atopology on a universe the notions of lower, upper, and rough approximationspaces can be introduced inheriting all the properties discussed in the general

A Hierarchical Lattice Closure Approach 369

covering context. In particular, the inclusion uγ(A) ⊂ u(op)γ (A) continues to hold

in this context. Moreover, the lower (resp., upper) approximation is a Kuratowskiinterior (resp., closure) operation.

3.3 Partitions of a Universe as Models of Halmos Closure

A partition of X is a (non necessarily finite) collection π = {Gi : i ∈ I} ofnonempty subsets of X such that the following hold: covering condition: X =∪{Gi ∈ π : i ∈ I} ; disjointness condition: for any pair Gh, Gk of differentelements, Gh �= Gk, of the partition π it is Gh ∩ Gk = ∅. The usual approachto rough set theory as introduced by Pawlak [11,12,13] is based on a concretepartition space, that is a pair (X, π) consisting of a nonempty set X , the universeof discourse (with corresponding power set P(X), the collection of all subsets ofX , which are the approximable sets), and a partition π := {Gi ∈ P(X) : i ∈ I} ofX whose elements are the elementary sets . The partition π can be characterizedby the induced equivalence relation R ⊆ X×X , defined as (x, y) ∈ R iff ∃G ∈π : x, y ∈ G. In this case x, y are said to be indistinguishable with respect to Rand the equivalence relation R is called an indistinguishability relation.

Of course, a partition space (X, π) generates a topological space whose openbase is just the family π, i.e., B = π; the corresponding closed base is defined inthe usual way. A partition gives rise to a topology of clopen, denoted by Eπ(X),formally: Eπ(X) = Oπ(X) = Cπ(X). Thus Eπ(X) is the collection of all crisp sets(i.e., if in this model the collection of all approximable elements is Σ = P(X),then the corresponding collection of all crisp set is Σc = Eπ(X)).

Thus, the concrete rough approximation space generated by the partition πconsists of the structure Rπ := 〈P(X), Eπ(X), lπ, uπ〉 where: P(X) is the Booleanatomic (complete) lattice of all approximable subsets A of the universe X ; thetwo sets of lower crisp elements and of upper crisp elements coincide with theset of clopen (exact) elements Eπ(X); the map lπ : P(X) �→ Eπ(X) associateswith any subset A of X its lower approximation:

lπ(A) := ∪{O ∈ Eπ(X) : O ⊆ A} = ∪{G ∈ π : G ⊆ A} ; (5a)

the map uπ : P(X) �→ Eπ(X) associates with any subset A of X its upperapproximation

uπ(A) := ∩{C ∈ Eπ(X) : A ⊆ C} = ∪{H ∈ π : H ∩ A �= ∅} . (5b)

Note that for any subset A, the universe turns out to be the set theoreticunion of the mutually disjoint sets X = lπ(A) ∪ bπ(A) ∪ eπ(A), where bπ(A) :=uπ(A) \ lπ(A) is the boundary and eπ(A) := X \ uπ(A) is the exterior of A.

Let us stress that in the case of a covering approximation space, as gener-alization of the partition approximation space, the equations (4a) and (4b) aregeneralizations of the corresponding partition versions (5a) and (5b). Further,the lower (resp., upper) approximation lπ (resp., uπ) is a Halmos interior (resp.,closure) operation.

370 G. Cattaneo and D. Ciucci

4 ConclusionIn the present work, we discussed three different notions of closure on a De Mor-gan lattice: Tarski, Kuratowski and Halmos, giving rise to a hierarchy of operators.For each of them a model of sets is discussed: coverings, topologies and partitions,respectively. Further, to any closure operator it is possible to associate a dual in-terior operator such that the pair interior-closure is a rough approximation.

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