Topologies associated to (dual) fuzzy approximation operators on fuzzy finite state automata

1 Introduction

The notion of fuzzy sets was introduced by Zadeh [11] in 1965. After that many researchers worked in this field and developed it. One of the applications of fuzzy sets is in automata theory. Wee in [10] introduced and studied fuzzy automata, and later in [2] the authors introduced a generally formal defenition. Also, two fuzzy topologies on the state set of a fuzzy finite state automata based on fuzzy closure operators were introduced by [1, 9]. In this paper at first we introduce dual fuzzy approximation operator C ′ ¯, then we show that this operator induces a topology τ ′ ¯, which contains the members exactly complementary of those in τ ¯ [9]. Finally, we introduce two (interior) fuzzy approximation operators and obtain their associated topologies.

2 Preliminaries

This section is devoted to recall some concepts and notions from fuzzy topology and fuzzy automata. The presented results are a summary of those appearing in several books and papers [1, 3-6]. Along this paper we denote by F (X) the set of all fuzzy subsets of X.

Definition 2.1. [5] Let t ∈ [0, 1] and x ∈ X. Then we name x _t as fuzzy point if x t ( a ) = { t , i f a = x , 0 , o t h e r w i s e ..

Definition 2.2. [9] A pair (X, R) is called an approximation space if X is a set and R is a binary relation on X.

Definition 2.3. [5] A fuzzy finite state machine (denoted ffsm) is a triple M = (Q, ∑, μ) such that Q and ∑ are nonempty finite sets of states and inputs and μ is a fuzzy subset of Q × ∑ × Q.

Definition 2.4. [1] Let M = (Q, ∑, μ) be a ffsm. Let δ be a fuzzy subset of Q. Then δ is called a subsystem of M if, δ ( q ) ≥ δ ( p ) ∧ μ ( p , a , q ) for all p , q ∈ Q , a ∈ ∑ . Definition 2.5. [1] Let M = (Q, ∑, μ) be a ffsm and let δ be a fuzzy subset of Q. Then δ is called a strong subsystem of M if, μ ( p , a , q ) > 0 ⇒ δ ( p ) ≤ δ ( q ) for all p , q ∈ Q , a ∈ ∑ . Definition 2.6. [1] Let ∑^* denotes the set of all words of elements of ∑ included empty word Λ. Then μ^* is defined as follows: μ * : Q × ∑ * × Q ⟶ [ 0 , 1 ] , μ * ( q , Λ , p ) = { 1 , q = p ; 0 , q ≠ p . μ * ( q , xa , p ) = ∨ { μ * ( q , x , r ) ∧ μ ( r , a , p ) , r ∈ Q } , ∀ p , q ∈ Q , x ∈ ∑ * , a ∈ ∑ . Definition 2.7. [5] A fuzzy closure (interior) operator on a set X is a mapping satisfying the following conditions:

ψ (0) =0, (I (1) =1) ,

Definition 2.8. [9] Assume that ▵ is a fuzzy topology on X and α ∈ [0, 1]. Then i _a  (▵) = {λ^-1 (α, 1] | λ ∈ ▵} is a topology, reffered to as α-level topology of ▵.

Definition 2.9. [9] Assume that {i _α  (▵)} _α∈[0,1) is family of α-level topologies of ▵. Then sup(i _α  (▵)) denoted i (▵) is a topology, called the topological modification of the fuzzy topology ▵.

Theorem 2.10. [6] Suppose that we have a set X. Then C is a closure operator iff I defined as I (A) = (C (A ^c ))  ^c for all A ∈ p (X) is an interior operator.

Definition 2.11. [5] Let M _i  = (Q _i , ∑ _i , μ _i ) be a ffsm for i = 1, 2. A pair of mappings (f, g), where f : Q₁ ⟶ Q₂, g : ∑₁ ⟶ ∑₂, is called a homomorphism of M₁ to M₂ if μ₂ (f (p) , g (x) , f (q)) ≥ μ₁ (p, x, q), for all p, q ∈ Q, x ∈ ∑₁. In the above definition if ∑₁ = ∑₂ and g is the identity map of ∑₁, then we say that f is a homomorphism of M₁ to M₂.

Definition 2.12. [5] Suppose that M is a ffsm. Then successor operator, denoted as S is defined as follows,

S : P (Q) ⟶ P (Q)

S (T) = {p ∈ Q|  ∃ (x, q) ∈ ∑^* × T,  μ (q, x, p) >0} . Moreover, if T = {q}, then we show S ({q}) by S (q) and it is trivial that S (T) = ⋃ _q∈TS (q).

Theorem 2.13. [5] Let M be a ffsm and S : P (Q) ⟶ P (Q) such that S is a successor operator. Then for all A and any family {A _i } _i∈I in P (Q) it holds:

S (∅) = ∅ ,

Similarly Theorems 2.13 is true for source operator, defined as follows,

σ : P (Q) ⟶ P (Q) ,

σ (T) = {p ∈ Q| ∃ (x, q) ∈ ∑^*  × T, μ^* (p, x, q) >0} .

Moreover, it is trivial p ∈ σ (q) ⇔ q ∈ S (p).

Definition 2.14. [1] Let M = (Q, ∑, μ) be a ffsm. A (fuzzy) closure operator ψ on the state set of M induce (fuzzy) topologies on Q to be denoted by δ (ψ) and is as follows: δ ( ψ ) = { A ∈ P ( Q ) | ψ ( A c ) = A c } , δ ( ψ ) = { λ ∈ F ( Q ) | ψ ( λ c ) = λ c } . Definition 2.15. [1] Let M = (Q, ∑, μ) be a ffsm. A (fuzzy) interior operator I on the state set of M induce (fuzzy) topologies on Q to be denoted by δ (I) and is as follows: δ ( I ) = { A ∈ P ( Q ) | I ( A c ) = A c } , δ ( I ) = { λ ∈ F ( Q ) | I ( λ c ) = λ c } . Definition 2.16. [1] Suppose that δ (ψ) is a topology associated with a closure operator ψ. Then δ (ψ) is saturated if ⋂_i∈IA _i  ∈ δ (ψ) for all A _i  ∈ δ (ψ) where i ∈ I. Moreover, this is equivalent to say that ψ (⋃ _i∈IA _i ) = ⋃ _i∈Iψ (A _i ) for any family {A _i } _i∈I in P (X).

Similarly Definition 2.15 and Theorem 2.13 holds, with some proper changes for closure operator and also (fuzzy) interior operator. Let M = (Q, ∑, μ) be a ffsm. The source and the successor operators induce two saturated topologies on Q to be denoted, respectively, as T^* (Q) and T (Q), as follows: T * ( Q ) = { A ∈ P ( Q ) | σ ( A c ) = A c } , T ( Q ) = { A ∈ P ( Q ) | S ( A c ) = A c } . Definition 2.17. [4] Let I be an indexed set. Let T be a family of fuzzy sets in F (X). Then T is called a fuzzy Lowen topology on X if it holds: i ) α ∈ T , ∀ α ∈ [ 0 , 1 ] , ii ) μ ∧ ν ∈ T , ∀ μ , ν ∈ T , iii ) ⋁ i ∈ I μ i ∈ T , ∀ { μ i } ⊆ T . Definition 2.18. [5] Let M be a ffsm and p, q ∈ Q and T ⊆ Q. M is said to has exchange property if assuming p ∈ S (T ∪ {q}), then q ∈ S (T ∪ {p}).

Theorem 2.19. [5] If M is a ffsm, then the followings are equivalent

1) M has exchange property.

2) q ∈ σ (p) ⇔ p ∈ σ (q) for all p, q ∈ Q.

3 Dual fuzzy approximation operator

In this section we consider a dual approximation operator C ′ ¯ on a set X to induce a fuzzy topology on X. Then we obtain some main properties of this topology in particular relevant to fffsm.

Definition 3.1. Let (X, R) be an approximation space. Then we call C ′ ¯ : F ( X ) ⟶ F ( X ) defined as, C ′ ¯ ( λ ) ( x ) = ⋁ { λ ( y ) | x ∈ R ( y ) } , λ ∈ F ( x ) , x ∈ X, the dual fuzzy operator on X (induced by R).

Theorem 3.2. A relation R on a set X is reflexive and transitive iff dual fuzzy operator C ′ ¯ is saturated fuzzy closure operator on X.

Proof. It is trivial by Theorem 2.3 [7], with some proper changes.

Theorem 3.3. Let R be a reflexive and transitive relation on X. Then dual fuzzy approximation operator C ′ ¯ induces a fuzzy topology on X. We denote it by τ ′ ¯ ( X ).

Proof. From Theorem 3.2, dual fuzzy approximation operator C ′ ¯ is saturated fuzzy closure operator on F (X). Now let τ ′ ¯ ( X ) = { λ ∈ F ( X ) | C ′ ¯ ( λ C ) = λ C }.

Definition 3.4. Let C ′ ¯ be the dual fuzzy approximation operator. We call S ′ ¯ : P ( X ) ⟶ P ( X ) the associated dual approximation operator and define it by

S ′ ¯ ( A ) = ( C ′ ¯ ( χ A ) ) - 1 ( 1 ) = { x ∈ X | C ′ ¯ ( χ A ) ) ( x ) = 1 } , A ∈ P ( X ) .

Corollary 3.5. Let S ′ ¯ be associated dual approximation operator. Then it holds S ′ ¯ ( A ) = { x ∈ X | x ∈ R ( A ) }, whereas R (A) = ⋃ _x∈AR (x).

Proof. By definition S ′ ¯ ( A ), we have S ′ ¯ ( A ) = ( C ′ ¯ ( χ A ) ) - 1 ( 1 ) = { x ∈ X | C ′ ¯ ( χ A ) ( x ) = 1 } = { x ∈ X | ∨ { χ A ( y ) | x ∈ R ( y ) } = 1 } = { x ∈ X | ∃ y ∈ A , x ∈ R ( y ) } = { x ∈ X | x ∈ R ( A ) } .

Theorem 3.6. A relation R on X is reflexive and transitive iff (associated) dual approximation operator S ′ ¯ is a saturated closure operator on X.

Proof. It is trivial by Theorem 2.1 [7] with some proper changes.

Theorem 3.7. Let R be a reflexive and transitive relation on X. Then approximation operator S ′ ¯ induces a saturated ordinary topology on X, denoted as T ′ ¯ ( X ).

Proof. By Theorem 3.6, dual approximation operator S ′ ¯ is a saturated fuzzy closure operate on P (X). Now let T ′ ¯ ( X ) = { A ⊆ X | S ′ ¯ ( A C ) = A C } .

Theorem 3.8. Let T ′ ¯ ( X ) and τ ′ ¯ ( X ) be the topologies mentioned in Theorems 3.3 and 3.7. Then i α ( τ ′ ¯ ( X ) ) = T ′ ¯ ( X ), whereas α ∈ [0, 1), i ( τ ′ ¯ ( X ) ) = T ′ ¯ ( X ).

Proof. Let α ∈ [0, 1). Then i α ( τ ′ ¯ ( X ) ) = { λ - 1 ( α , 1 ] | λ ∈ τ ′ ¯ ( X ) } = { λ - 1 ( α , 1 ] | C ′ ¯ ( 1 - λ ) = 1 - λ , λ ∈ F ( X ) } , on the other hand λ^-1 (α, 1] = (1 - λ) ^-1 [0, 1 - α) . So, i α ( τ ′ ¯ ( X ) ) = { ( 1 - λ ) - 1 [ 0 , 1 - α ) | C ′ ¯ ( 1 - λ ) = 1 - λ , λ ∈ F ( X ) } and T ′ ¯ ( X ) = { A ∈ P ( X ) | S ′ ¯ ( X - A ) = X - A } .

Now we set ( 1 - λ ) - 1 [ 0 , 1 - α ) = A ∈ i α ( τ ′ ¯ ( X ) ) .

We conclude that C ′ ¯ ( 1 - λ ) = 1 - λ . Conversely we have S ′ ¯ ( X - A ) = X - A . So X - A ⊆ S ′ ¯ ( X - A ) Now suppose that x ∈ S ′ ¯ ( X - A ). Hence x ∈ R (X - A). So there is y ∈ X - A such that x ∈ R′ (y) . On the other hands, y ∈ X - A implies that (1 - λ) (y) ≥1 - α. So, ( 1 - λ ) ( x ) = C ′ ¯ ( 1 - λ ) ( x ) = ∨ { ( 1 - λ ) ( y ) | x ∈ R ( y ) } ≥ 1 - α , i.e., x ∉ (1 - λ) ^-1 [0, 1 - α) = A . Thus x ∈ X - A. Now assume that A ∈ T ′ ¯ ( X ). Hence S ′ ¯ ( X - A ) = X - A . Conversely, let A ∈ T ′ ¯ ( X ). Now it suffices to show,

A = χ X - A - 1 [ 0 , 1 - α ),

Assertion (1) is trivial. Let x ∈ X. Then, if C ′ ¯ ( χ X - A ) = 0, we have C ′ ¯ ( χ X - A ) = ∨ { χ X - A ( y ) | x ∈ R ( y ) }. So, χ_X-A (y) =0, for all y such that x ∈ R (y). Thus χ_X-A (x) =0.

Now, if C ′ ¯ ( χ X - A ) > 0, then there is y ∈ X such that x ∈ R (y), χ_X-A (x) >0. Thus χ_X-A (y) =1 i.e., y ∈ X - A. Therefore, R (y) ⊆ R (X - A). So x ∈ R (X - A). That is x ∈ S ′ ¯ ( X - A ) = X - A. So χ_X-A (x) =1 and so C ′ ¯ ( χ X - A ) ( x ) ≤ χ X - A ( x ) . Thus we have T ′ ¯ ( X ) ⊆ τ ′ ¯ ( X ).

Lemma 3.9. Let R be a relation on the state set of a ffsm M = (Q, ∑, μ) such that pRq ⇔ p ∈ σ (q) for all p, q ∈ Q. Then R is reflexive and transitive.

Lemma 3.10. Let M = (Q, ∑, μ) be a ffsm,

(1) Dual fuzzy approximation operator C ′ ¯ defined as, C ′ ¯ ( λ ) ( q ) = ⋁ { λ ( p ) | q ∈ σ ( p ) }, for all λ ∈ F (Q), p ∈ Q, is a saturated fuzzy closure operator on Q.

(2) For all A ∈ P (Q), associated dual approximation operator S ′ ¯, defined as S ′ ¯ ( A ) = { q ∈ Q | q ∈ σ ( A ) }, is a saturated closure operator on Q. Note that clearly S ′ ¯ ( A ) = σ ( A ) .

Proof. By Lemma 3.9, Theorems 3.2 and 3.6, the assertions hold.

Remark 3.11. Let M = (Q, ∑, μ) be a ffsm. Then C ′ ¯ induces a fuzzy topology τ ′ ¯ ( Q ). So using Lemma 3.10 and noticing S ′ ¯ ( A ) = σ ( A ), S ′ ¯ induces T (Q).

Theorem 3.12. Let M = (Q, ∑, μ) be a ffsm and λ ∈ F (Q). Then λ is a strong fuzzy subsystem of M iff λ ∈ τ ′ ¯ ( Q ).

Proof. (⇒). Let λ be a strong fuzzy subsystem of M. We show that C ′ ¯ ( λ C ) ≤ λ C, for all q ∈ Q if q ∈ σ (p), then λ (q) ≥ λ (p), i.e., C ′ ¯ ( λ C ) ≤ λ C.

(⇐). Let λ ∈ τ ′ ¯ ( Q ). Then C ′ ¯ ( λ C ) = λ C. Also, suppose q ∈ σ (p), p, q ∈ Q. Since λ C ( q ) = C ′ ¯ ( λ C ) ( q ) = ∨ { λ C ( p ) | q ∈ σ ( p ) } , we have λ ^C  (q) ≥ λ ^C  (p). Thus λ (q) ≤ λ (p) and so λ is a strong fuzzy subsystem.

Corollary 3.13. Let M = (Q, ∑, μ) be a ffsm and λ ∈ F (Q). Then λ ∈ τ ′ ¯ ( Q ) iff λ c ∈ τ ¯ ( Q ).

Proof. The result is an immediate consequences of Theorem 3.3 and Theorem 4.4 [7].

Corollary 3.14. If τ ′ ¯ ( Q ) and T (Q) be the topologies mentioned in Corollary 3.12, then i ( τ ′ ¯ ( Q ) ) = T ( Q ).

Corollary 3.15. Let f : (Q, ∑, μ) ⟶ (R, ∑, δ) be a homomorphism. Then f : ( Q , τ ′ ¯ ( Q ) ) ⟶ ( R , τ ′ ¯ ( R ) ) is fuzzy continuous.

Proof. In the way similar to proof of Theorem 4.6 [7], with some manipulations.

4 Interior fuzzy approximation space

In this section we consider an approximation space (X, R) to define an interior operator and dual fuzzy approximation operator on X. Then we use these operations to induces a Lowen saturated fuzzy topology on X, we denote it by τ _ ( X ).

Theorem 4.1. [3] Let X be an arbitrary set. Then C _ : P ( X ) ⟶ P ( X ) such that C _ ( λ ) = ( C ¯ ( λ c ) ) c for all λ ∈ P (X), is an interior operator on X iff C ¯ is a closure operator.

Definition 4.2. Let (X, R) be an approximation space. Then we define dual fuzzy approximation operator C _ as follows: C _ : F ( X ) ⟶ F ( X ) , C _ ( λ ) ( x ) = ⋀ { λ ( y ) | y ∈ R ( x ) } , λ ∈ F ( X ) , x ∈ X .

A binary relation R on X is reflexive and transitive iff C _ is a saturated fuzzy interior operator on X. Let R be a reflexive and transitive relation on X. Then fuzzy interior operator C _ induces a Lowen saturated fuzzy topology on X, we denote it by τ _ ( X ).

Definition 4.3. Let C _ be a fuzzy interior approximation operator. Then we define associated dual approximation operator S _ : P ( X ) ⟶ P ( X ) as follows, S _ ( A ) = ( C _ ( χ A ) ) - 1 ( 1 ) , A ∈ P ( X ) .

Let S _ be dual approximation operator of S ¯. Then we have, S _ ( A ) = { x ∈ X | R ( x ) ⊆ A }.

Theorem 4.4. A relation R on X is reflexive and transitive iff the operator S _ is a saturated interior operator on X.

Proof. It is easily seen that S _ = ( S ¯ ( A C ) ) C.

Theorem 4.5. Let R be a reflexive and transitive relation on X. Then S _ induces a saturated ordinary topology on X and we denote it as T _ ( X ).

Proof. Assuming T _ ( X ) = { A ∈ P ( X ) | S _ ( A ) = A } the assertion easily holds.

Theorem 4.6. Let k ¯ and k _ be dual (fuzzy) approximation operators in Definition 3.1 and Definition 4.2. Assume Ω _ and Ω ¯ are (fuzzy) topologies, associated to k _ and k ¯, respectively, then A ∈ Ω ¯ iff A c ∈ Ω _.

Proof. Let A ∈ Ω ¯. Then k ¯ ( A C ) = A C. Consequently, ( k ¯ ( A C ) ) C = A. Thus by definition of Ω _ we have k _ ( A ) = A . Since k _ is a (fuzzy) interior operator, then A ∈ Ω ¯.

Remark 4.7. It is easily seen that S ¯ ( A ) = ( S _ ( A c ) ) c, C ¯ ( λ ) = ( C _ ( λ c ) ) c. So C _ ( S _ ) is (fuzzy) interior operator associated to C ¯ ( S ¯ ). Thus by Theorem 4.6 we have ( λ ∈ τ ¯ ( X ) ) A ∈ T ¯ ( X ) iff ( λ c ∈ τ _ ( X ) ) A c ∈ T _ ( X ).

Theorem 4.8. For all α ∈ [0, 1) we have i α ( τ _ ( X ) = T _ ( X ). i ( τ _ ( X ) = T _ ( X ).

Proof. Let α ∈ [0, 1). Then we have, i α ( τ _ ( X ) ) = { λ - 1 ( α , 1 ] | λ ∈ τ _ ( X ) } = { λ - 1 ( α , 1 ] | C _ ( 1 - λ ) = 1 - λ , λ ∈ F ( X ) } , on the other hand λ^-1 (α, 1] = (1 - λ) ^-1 [0, 1 - α) . So, i α ( τ _ ( X ) ) = { ( 1 - λ ) - 1 ( 0 , 1 - α ] | C _ ( 1 - λ ) = 1 - λ , λ ∈ F ( X ) }

Now let ( 1 - λ ) - 1 [ 0 , 1 - α ) = A ∈ i α ( τ _ ( X ) ) .

We show that A ∈ T _ ( X ). To do this it suffices to show that S _ ( X - A ) = X - A . Since S _ is a fuzzy interior operator S _ ( X - A ) ⊆ X - A. Let x ∈ X - A. Hence x ∉ A. Thus x ∉ (1 - λ) ^-1 [0, 1 - α). Consequently, ( 1 - λ ) ( x ) = C _ ( 1 - λ ) ( x ) = ∧ { ( 1 - λ ) ( y ) | y ∈ R ( x ) } ≥ 1 - α , thus (1 - λ) (y) ≥1 - α, for all y ∈ R (x). So y ∈ X - A, for all y ∈ R (x). It follows that R (x) ⊆ X - A.

Now suppose that A ∈ T _ ( X ). We have S _ ( X - A ) = X - A. We show that

A = χ X - A - 1 [ 0 , 1 - α ),

Assertion (1) is trivial. Let us prove assertion (2).

Let x ∈ X. Then, if χ_X-A (x) =0, thenχ X - A ≤ ( _ χ X - A ) ( x ). But if χ_X-A (x) >0 holds since χ_X-A = 1, it follows that x ∈ X - A. Hence x ∈ S _ ( X - A ) and R (x) ⊆ X - A, i.e., ∧{1_X-A (y) |y ∈ R (x)} =1. Therefore, C _ ( χ X - A ) ( x ) = 1. Consequently in both cases we have χ X - A ≤ C _ ( χ X - A ). On the other hand C _ ( χ X - A ) ≤ χ X - A and hence C _ ( χ X - A ) = χ X - A. So A ∈ i α ( τ _ ( X ) ). Therefore, T _ ( X ) ⊆ τ _ ( X ).

Lemma 4.9. Let M = (Q, ∑, μ) be a ffsm. We have:

Dual fuzzy approximation operator C _ which is defined as C _ ( λ ) ( q ) = ⋀ { λ ( p ) : p ∈ σ ( q ) } for all λ ∈ F ( Q ) , q ∈ Q , is a saturated fuzzy interior operator on Q.

Remark 4.10. Let M = (Q, ∑, μ) be a ffsm. Then C _ induces a saturated interior operator, denoted by τ _ ( Q ) and also it is obvious that S _ induces saturated topology T (Q). i ( τ _ ( Q ) ) = T ( Q ).

Theorem 4.11. Let f : (Q, ∑, μ) ⟶ (R, ∑, δ) be a homomorphism. Then f : ( Q , τ _ ( Q ) ) ⟶ ( R , τ _ ( R ) ) is fuzzy continuous.

Proof. It is similar to Theorem 4.6 [7] with some proper changes.

Definition 4.12. Let (X, R) be an approximation space. We define dual fuzzy approximation operator C _ ′ as follows: C _ ′ : F ( X ) ⟶ F ( X ) , C _ ′ ( λ ) ( x ) = ⋀ { λ ( y ) | x ∈ R ( y ) } , λ ∈ F ( X ) , x ∈ X .

Corollary 4.13. A relation R on X is reflexive and transitive iff C _ ′ is a saturated fuzzy interior operator on X.

Proof. Let R^* be a relation on X such that y ∈ R^* (x) iff x ∈ R (y). Then it is seen that R^* is reflexive and transitive iff R is reflexive and transitive, in other word, by Definition 4.2, C ′ _ is a saturated fuzzy interior operator on X.

Corollary 4.14. Let R be a reflexive and transitive relation on X. Then C _ ′ induces a Lowen saturated fuzzy topology on X and we denote it as τ _ ′ ( X ).

Definition 4.15. Let C _ ′ be a fuzzy approximation interior operator. Then we define approximation operator S _ ′, as follows: S _ ′ : P ( X ) ⟶ P ( X ) , S _ ′ ( A ) = ( C _ ′ ( χ A ) ) - 1 ( 1 ) , A ∈ P ( X ) , and then S _ ′ ( A ) = { x ∈ X | x ∉ R ( A c ) } .

A relation R on X is reflexive and transitive iff S _ ′ is a saturated interior operator on X.

Theorem 4.16. Let R be a reflexive and transitive relation on X. Then S _ ′ induces a saturated ordinary topology on X. We denote it as T _ ′ ( X ).

Similarly all items from 4 to 4 are true for T _ ′.

Corollary 4.17. Let M = (Q, ∑, μ) has exchange property. Then if R be a binary relation on state set of M such that pRq ⇔ p ∈ σ (q), for all p, q ∈ Q, then we have, ( 1 ) S ′ ¯ ( A ) = ⋃ q ∈ Q { σ ( q ) | σ ( q ) ∩ A ≠ ∅ } , ( 2 ) S ′ _ ( A ) = ⋃ q ∈ Q { σ ( q ) | σ ( q ) ⊆ A } .

Proof. Since M has exchange property it follows p ∈ σ (q) ⇔ q ∈ σ (p). Hence the relation R such that pRq ⇔ p ∈ σ (q), for all p, q ∈ Q is an equivalence relation on Q. Now let σ (p) = {q ∈ Q| (p, q) ∈ R} and N = ∪ _q∈Q {σ (q) |σ (q) ∩ A ≠ φ}. By definition of S ′ ¯ ( A ) we have, S ′ ¯ ( A ) = { p ∈ Q | ∃ q ∈ A , p ∈ R ( q ) } = { p ∈ Q | p ∈ R ( A ) } = { p ∈ Q | p ∈ σ ( A ) }. If p ∈ S ′ ¯ ( A ), then there is q ∈ A such that p ∈ σ (q). Hence p, q are in the same equivalence class σ (q) and also σ (q) ∩ A ≠ φ. So p ∈ ∪ _q∈Q {σ (q) |σ (q) ∩ A ≠ φ}. It follows that S ′ ¯ ( A ) ⊆ N. Now let p ∈ N. Then there is r ∈ Q such that σ (r) ∩ A ≠ φ. So there is q ∈ A such that q ∈ σ (r). This follows that p, q are in the same equivalence class. Thus, p ∈ σ (q), consequently p ∈ S ′ ¯ ( A ),i.e., N ⊆ S ′ ¯ ( A ) and this complete the proof.