Closure and Interior Systems

The notions of closure system and interior system introduced in this section are significant in algebra and topology and have applications in the study of frequent item sets in data mining.

Definition 1.166 Let S be a set. A closure system on S is a collectionCof subsets of S that satisfies the following conditions:

(i) S∈Cand

(ii) for every collectionD⊆C, we have D∈C.

Example 1.167 LetCbe the collection of all intervals[a,b] = {x∈R | axb} with a,b ∈ Rand a b together with the empty set and the setR. Note that

⎜C=R∈C, so the first condition of Definition 1.166 is satisfied.

Let D be a nonempty subcollection ofC. If ∅ ∈ D, then

D = ∅ ∈ C. If D = {R}, then

D = R ∈ C. Therefore, we need to consider only the case when D = {[a_i,b_i] | i ∈ I}. Then,

D = ∅unless a = sup{a_i | i ∈ I}and b=inf{b_i | i∈I}both exist and ab, in which case

D= [a,b]. Thus,Cis a closure system.

Example 1.168 Let A = (A,I)be an algebra and let S(A)be the collection of subalgebras ofA,S(A)= {(A_i,I) | i ∈ I}. The collectionS= {A_i | i ∈ I}is a closure system. It is clear that we have S ∈ S. Also, if{A_i | i ∈ J}is a family of subalgebras, then

i∈JA_iis a subalgebra ofA.

Many classes of relations define useful closure systems.

1.8 Closure and Interior Systems 43 Theorem 1.169 Let S be a set and let REFL(S),SYMM₍S)andTRAN₍S)be the sets of reflexive relations, the set of symmetric relations, and the set of transitive relations on S, respectively. Then, REFL(S),SYMM₍S)andTRAN₍S)are closure systems on S.

Proof Note that S×S is a reflexive, symmetric, and transitive relation on S. There-fore,⎜ andSYMM₍S)are also closure systems.

Theorem 1.170 The set of equivalences on S, EQ(S), is a closure system.

Proof The relationθ_S=S×S, is clearly an equivalence relation as we have seen in the proof of Theorem 1.169. Thus,⎜

for U ∈P(R). We leave to the reader the verification that K is a closure operator.

Closure operators induce closure systems, as shown by the next lemma.

Lemma 1.173 Let K: P(S)−→P(S)be a closure operator. Define the family of

Note thatCK, as defined in Lemma 1.173, equals the range of K. Indeed, if L∈ Ran(K), then L=K(H)for some H∈P(S), so K(L)=K(K(H))=K(H)=L, which shows that L∈CK. The reverse inclusion is obvious.

We refer to the sets inCKas the K-closed subsets of S.

In the reverse direction from Lemma 1.173, we show that every closure system generates a closure operator.

Lemma 1.174 Let Cbe a closure system on the set S. Define the mapping K_C : P(S)−→P(S)by K_C(H)=

{L ∈ C | H ⊆L}. Then, K_Cis a closure operator on the set S.

Proof Note that the collection{L ∈C | H ⊆L}is not empty since it contains at least S, so K_C(H)is defined and is clearly the smallest element ofCthat contains H.

Also, by the definition of K_C(H), it follows immediately that H ⊆K_C(H)for every H∈P(S).

Suppose that H₁,H₂∈P(S)are such that H₁⊆H₂. Since {L∈C | H₂⊆L} ⊆ {L∈C | H₁⊆L},

we have ⎟

{L∈C | H1⊆L} ⊆⎟

{L∈C | H2⊆L}, so K_C(H1)⊆K_C(H2).

We have K_C(H)∈Cfor every H∈P(S)becauseCis a closure system. Therefore, K_C(H) ∈ {L ∈ C | K_C(H) ⊆ L}, so K_C(K_C(H)) ⊆ K_C(H). Since the reverse inclusion clearly holds, we obtain K_C(K_C(H))=K_C(H).

Definition 1.175 LetCbe a closure system on a set S and let T be a subset of S. The C-set generated by T is the set K_C(T).

Note that K_C(T)is the least set inCthat includes T .

Theorem 1.176 Let S be a set. For every closure systemCon S, we haveC=CK_C. For every closure operator K on S, we have K=K_CK.

Proof LetCbe a closure system on S and let H⊆M. Then, we have the following equivalent statements:

1. H ∈CK_C. 2. K_C(H)=H.

3. H ∈C.

The equivalence between (2) and (3) follows from the fact that K_C(H)is the smallest element ofCthat contains H.

Conversely, let K be a closure operator on S. To prove the equality of K and K_CK, consider the following list of equal sets, where H ⊆S:

1.8 Closure and Interior Systems 45 1. K_CK(H).

2.

{L∈CK | H⊆L}.

3.

{L∈P(S) | H ⊆L=K(L)}. 4. K(H).

We need to justify only the equality of the last two members of the list. Since H ⊆ K(H) = K(K(H)), we have K(H) ∈ {L ∈ P(S) | H ⊆ L = K(L)}. Thus, {L∈P(S) | H⊆L=K(L)} ⊆K(H). To prove the reverse inclusion, note that for every L∈ {L∈P(S) | H⊆L=K(L)}, we have H⊆L, so K(H)⊆K(L)=L.

Therefore, K(H)⊆

{L∈P(S) | H⊆L=K(L)}.

Theorem 1.176 shows the existence of a natural bijection between the set of closure operators on a set S and the set of closure systems on S.

Definition 1.177 LetCbe a closure system on a set S and let T be a subset of S. The C-closure of the set T is the set K_C(T).

As we observed before, K_C(T)is the smallest element ofCthat contains T . Example 1.178 Let K be the closure operator given in Example 1.172. Since the closure systemCK equals the range of K, it follows that the members ofCK, the K-closed sets, are∅,R, and all closed intervals[a,b]with a b. Thus,CKis the closure systemCintroduced in Example 1.167. Therefore, K andCcorrespond to each other under the bijection of Theorem 1.176.

For a relationρ, on S defineρ⁺ as K_TRAN(S)(ρ). The relationρ⁺ is called the transitive closure ofρand is the least transitive relation containingρ.

Theorem 1.179 Letρbe a relation on a set S. We have ρ⁺=

{ρⁿ | n∈Nand n1}.

Proof Letτbe the relation⎜

{ρⁿ | n∈Nand n1}. We claim thatτis transitive.

Indeed, let(x,z), (z,y) ∈ τ. There exist p,q ∈ N, p,q 1 such that(x,z)∈ ρ^p and(z,y)∈ ρ^q. Therefore,(x,y) ∈ ρ^pρ^q = ρ^p+q ⊆ρ⁺, which shows thatρ⁺ is transitive. The definition of ρ⁺ implies that ifσis a transitive relation such that ρ⊆σ, thenρ⁺⊆σ. Therefore,ρ⁺⊆τ.

Conversely, sinceρ⊆ρ⁺we haveρⁿ⊆(ρ⁺)ⁿfor every n∈N. The transitivity of ρ⁺implies that(ρ⁺)ⁿ⊆ρ⁺, which impliesρⁿ⊆ρ⁺for every n1. Consequently, τ =⎜

{ρⁿ | n∈Nand n1} ⊆ρ⁺. This proves the equality of the theorem.

It is easy to see that the set of all reflexive and transitive relations on a set S, REFTRAN₍S), is also a closure system on the set of relations on S.

For a relation ρon S, defineρ^∗ as K_REFTRAN(S)(ρ). The relationρ^∗ is called the transitive-reflexive closure ofρand is the least transitive and reflexive relation containingρ. We have the following analog of Theorem 1.179.

Theorem 1.180 Letρbe a relation on a set S. We have ρ^∗=

{ρⁿ | n∈N}.

Proof The argument is very similar to the proof of Theorem 1.179; we leave it to the reader.

Definition 1.181 Let S be a set and let F be a set of operations on S. A subset P of S is closed under F, or F-closed, if P is closed under f for every f ∈F; that is, for every operation f ∈F, if f is n-ary and p₀, . . . ,p_n−1∈P, then f(p₀, . . . ,p_n−1)∈P.

Note that S itself is closed under F. Further, if Cis a nonempty collection of F-closed subsets of S, then

Cis also F-closed.

Example 1.182 Let F be a set of operations on a set S. The collection of all F-closed subsets of a set S is a closure system.

Definition 1.183 An interior operator on a set S is a mapping I :P(S)−→P(S) that satisfies the following conditions:

(i) U⊇I(U)(contraction),

(ii) U⊇V implies I(U)⊇I(V)(monotonicity), and (iii) I(I(U))=I(U)(idempotency),

for U,V ∈P(S). Such a mapping is known as an interior operator on the set S.

Interior operators define certain collections of sets.

Definition 1.184 An interior system on a set S is a collectionIof subsets of S such that

(i) ∅ ∈Iand,

(ii) for every subcollectionDof Iwe have⎜ D∈I.

Theorem 1.185 Let I:P(S)−→P(S)be an interior operator. Define the family of setsII= {U∈P(S) | U=I(U)}. Then,IIis an interior system on S.

Conversely, ifIis an interior system on the set S, define the mapping I_I:P(S)−→

P(S)by I_I(U)=⎜

{V ∈I | V ⊆U}. Then, I_Iis an interior operator on the set S.

Moreover, for every interior systemIon S, we haveI =II_I. For every interior operator I on S, we have I=I_II.

Proof This statement follows by duality from Lemmas 1.173 and 1.174 and from Theorem 1.176.

We refer to the sets inIIas the I-open subsets of S.

Theorem 1.186 Let K:P(S)−→P(S)be a closure operator on the set S. Then, the mapping L:P(S)−→P(S)given by L(U)=S−K(S−U)for U∈P(S)is an interior operator on S.

1.8 Closure and Interior Systems 47 Proof Since S−U⊆K(S−U), it follows that L(U)⊆S−(S−U)=U, which proves property (i) of Definition 1.184.

Suppose that U ⊆ V , where U,V ∈ P(S). Then, we have S−V ⊆S−U, so K(S−V)⊆K(S−U)by the monotonicity of closure operators. Therefore,

L(U)=S−K(S−U)⊆S−K(S−V)=L(V), which proves the monotonicity of L.

Finally, observe that we have L(L(U))⊆L(U)because of the contraction prop-erty already proven for L. Thus, we need only show that L(U)⊆L(L(U))to prove the idempotency of L. This inclusion follows immediately from

L(L(U))=L(S−K(S−U))⊇L(S−(S−U))=L(U).

We can prove that if L is an interior operator on a set S, then K:P(S)−→P(S) defined as K(U)=S−L(S−U)for U ∈ P(S)is a closure operator on the same set.

In Chap.4, we extensively use closure and interior operators.