Catalyzed LM -G-filter spaces

1 Introduction

In 1977, Lowen [9] introduced the concept of filters in I ^X and called them prefilters. Later in 1999 Burton et al. [3] introduced the concept of generalized filters as a map from 2 ^X to I. Subsequently, H o ̈ hle and S ˇ ostak [4] developed the notions of L-filters and stratified L-filters on a complete quasi-monoidal lattice. Generalizing the theory of stratified L-filters by introducing stratification mapping, where L and M are frames, J a ̈ ger [5] developed the theory of stratified LM-filters in 2013. Recently, Abd El-Latif et al. [1] developed the idea of LM-double fuzzy filter spaces and studied some of their fundamental properties. Later, Ramadan et al. [11] studied the relations among L-fuzzy pre-proximity spaces, L-fuzzy filters and L-fuzzy grills in 2020.

When the notions of sum, subspace, product and quotient are defined naturally for LM-filters, it is found that the quotient and sum remain LM-filters, whereas the subspace and product are no longer LM-filters. Moreover, the set of all LM-filters on a set X do not form a lattice! This inspired the authors to engender the concept of LM-G-filters in [6] for which all the essential properties hold good. Further, some subcategories of LM-G, the category of LM-G-filter spaces have been identified by introducing the concepts of inspired LM-G-filter spaces and weakly inspired LM-G-filter spaces in [7].

Now this paper introduces the concept of catalyzed LM-G-filter spaces obtained by investigating more subcategories of LM-G. It is proved that ILM-G, the category of inspired LM-G-filter spaces, is isomorphic to CLM-G, the category of catalyzed LM-G-filter spaces. It is also proved that CLM-G is an isomorphism-closed, bireflective and bicoreflective full subcategory of WILM-G, the category of weakly inspired LM-G-filter spaces, and LM-G. The study also describes the application potential of catalyzed LM-G-filter spaces in mathematical modeling and decision making processes.

2 Preliminaries

Throughout this paper, X stands for a non empty set and L and M stand for completely distributive lattices with an order reversing involution. An element α ∈ L is called prime if α < 1 and ∀a, b ∈ L, α ≥ a ∧ b ⇒ α ≥ a or α ≥ b. An element α ∈ L is called co-prime if α > 0 and ∀a, b ∈ L, a ∨ b ≥ α ⇒ a ≥ α or b ≥ α. The set of all prime elements and co-prime elements in L are denoted by pr (L) and co-pr (L) respectively. For all α ∈ L,  α _X  : X → L is defined by α (x) = α for every x ∈ X. For the various notions of category theory, the readers may refer to [2, 10].

Definition 2.1. [8] Let L be a complete lattice. Define a relation ⪯ in L as follows: ∀a, b ∈ L, a ⪯ b if and only if ∀S ⊂ L, ⋁ S ≥ b ⇒ ∃ s ∈ S such that s ≥ a .  ∀ a ∈ L, denote β (a) = {b ∈ L : b ⪯ a} , β^* (a) = co-pr (L) ⋂ β (a). For every a ∈ L, D ⊂ β (a) is called a minimal set of a, if ⋁D = a.

Definition 2.2. [8] Let L be a complete lattice. Define a relation ≻ ¯ in L as follows: ∀ a , b ∈ L , a ≻ ¯ b if and only if ∀S ⊂ L, ⋀ S ≤ b ⇒ ∃ s ∈ S such that s ≤ a .  ∀ a ∈ L, denote β _ ( a ) = { b ∈ L : b ≻ ¯ a } , β _ * ( a ) = pr ( L ) ⋂ β _ ( a ) . For every a ∈ L , D ⊂ β _ ( a ) is called a maximal set of a, if ⋀D = a.

From [8] we know that ∀a ∈ L, β^* (a) is a minimal set of a and β _ * ( a ) is a maximal set of a. Hence co-pr (L) is a join generating set of L and pr (L) is a meet generating set of L.

Definition 2.3. [5] A mapping s : L → M with the properties (M1)  s (0 _L ) = 0 _M  ; (M2)  s (1 _L ) =1 _M and (M3)  s (α ∧ β) = s (α) ∧ s (β) for all α, β ∈ L is called a stratification mapping.

Definition 2.4. [6] An LM-G-filter on a set X is defined to be a mapping G : L ^X  → M satisfying:

G (1 _X ) =1;

If G₁ and G₂ are two LM-G-filters on X such that G₂ (A) ≥ G₁ (A) for all A ∈ L ^X , then we say (X, G₁) is weaker (coarser) than (X, G₂) and (X, G₂) is stronger (finer) than (X, G₁).

Remark 2.5. In addition to the above axioms, if G4 : G (0 _X ) =0 is also satisfied, then (X, G) becomes an LM-filter space.

Definition 2.6. [6] Let (X, G₁) and (Y, G₂) be LM-G-filter spaces. A map f^→ : L ^X  → L ^Y is called an LM-G-filter map if G₁ (f^← (B)) ≥ G₂ (B) ,   ∀ B ∈ L ^Y  .

M-fuzzifying G-filter maps in M-fuzzifying G-filter spaces are defined analogously.

Definition 2.7. [6] Let (X, G) be an LM-G-filter space and s : L → M be a stratification mapping. Then (X, G) is called s-stratified LM-G-filter space if G (α _X ) ≥ s (α) for all α ∈ L.

Notation 2.8. [6] Let LM-G denotes the category of LM-G-filter spaces where morphisms are LM-G-filter maps. M-FYG denotes the category of M-fuzzifying G-filter spaces and M-FYG(X) denotes the lattice of set of all M-fuzzifying G-filters on X. Let SLM-G denotes the category of s-stratified LM-G-filter spaces.

Theorem 2.9. [6] SLM-G is an isomorphism-closed bicoreflective full subcategory of LM-G.

Definition 2.10. [6] Let (X, G) be an LM-G-filter space and Y ⊆ X. Then the LM-G-filter, G| _Y defined on Y by (G| _Y ) (B) = ⋁ {G (A) |A ∈ L ^X , A| _Y  = B} for all B ∈ L ^Y is called the subspace of (X, G).

Notation 2.11. Throughout this paper, ^′ ⊓ ′ stands for finite intersection.

Definition 2.12. [6] Let {(X _j , G _j )} _j∈J be a family of LM-G-filter spaces, X = ∏ j ∈ J X j and p j : ∏ j ∈ J X j → X j be the projection map. Then the product of {(X _j , G _j )} _j∈J is defined as G ( A ) = ⋁ ⊓ i ∈ I B i ≤ A ⋀ i ∈ I ⋁ j ∈ J ⋁ { G j ( D ) ; D ∈ X j , p j ← ( D ) = B i } for all A ∈ L ^X and the product LM-G-filter space is denoted by ( X , ∏ j ∈ J G j ) .

Definition 2.13. [6] Let (X, G) be an LM-G-filter space and f : X → Y be a surjective mapping. Then the LM-G-filter, G/f^→ defined on Y by G/f^→ (B) = G (f^← (B))   ∀ B ∈ L ^Y is called quotient LM-G-filter of G with respect to f.

Definition 2.14. [6] Let {(X _j , G _j )} _j∈J be a family of LM-G-filter spaces, X j ′ s be pairwise disjoint and X = ⋃ _j∈JX _j . Then the LM-G-filter, oplus _j∈JG _j defined on X by oplus _j∈JG _j  (A) = ⋀ _j∈JG _j  (A| _{X j} )   ∀ A ∈ L ^X is called sum LM-G-filter of {G _j } _j∈J.

Definition 2.15. [7] For A ∈ L ^X , define p-set of A by δ p ( A ) = { x ∈ X ; p ≻ ¯ A ( x ) } c where p is prime in L and ()  ^c denotes set complement. Let (X, G) be an LM-G-filter space. If G (A) = ⋀ _p∈pr(L)G (1_{δ p (A)}) for all A ∈ L ^X , then (X, G) is called an inspired LM-G-filter space. If G (A) ≤ ⋀ _p∈pr(L)G (1_{δ p (A)}) for all A ∈ L ^X , then (X, G) is called weakly inspired LM-G-filter space.

Notation 2.16. [7] Let WILM-G denotes the category of weakly inspired LM-G-filter spaces and ILM-G denotes the category of inspired LM-G-filter spaces.

Theorem 2.17. [7] Let (X, H) be an M-fuzzifying G-filter space. Then ζ (H) : L ^X  → M defined by ζ (H) (A) = ⋀ _p∈pr(L)H (δ _p  (A)) for all A ∈ L ^X is an LM-G-filter on X and ζ is a functor from M-FYG to LM-G.

Theorem 2.18. [7] Let G be an LM-G-filter on X. Then η (G) :2 ^X  → M defined by η (G) (U) = ⋁ _{⊓i∈IU i =U} ⋀ _i∈I ⋁ _p∈pr(L) ⋁ {G (A) ; A ∈ L ^X , δ _p  (A) = U _i } for all U ∈ 2 ^X is an M-fuzzifying G-filter on X and η is a functor from LM-G to M-FYG.

The functors η and ζ are related by the following theorem.

Theorem 2.19. [7]

η o ζ = id_M-FYG .

3 Catalyzed LM-G-filter spaces

Introducing the concept of catalyzed LM-G-filter spaces this section proves that CLM-G, the category of catalyzed LM-G-filter spaces, is isomorphic to ILM-G, the category of inspired LM-G-filter spaces.

Notation 3.1. For A ∈ L ^X , denote the 1-set of A by δ₁ (A) = {x ∈ X ; A (x) =1}.

Lemma 3.2. The following statements on 1-set are valid.

δ₁ (⋀ _λ∈ΛA _λ ) = ⋂ _λ∈Λδ₁ (A _λ ) for all {A _λ } _λ∈Λ ⊆ L ^X ;

The following theorem which associates an LM-G-filter space to a given M-fuzzifying G-filter space is an easy consequence of Lemma 3.2.

Theorem 3.3. Let (X, H) be an M-fuzzifying G-filter space and let ε (H) : L ^X  → M be defined by ε (H) (A) = H (δ₁ (A)) for all A ∈ L ^X  . Then ε (H) is an LM-G-filter on X .

Corollary 3.4. ε (H) (α _X  ∨ 1 _U ) = H (U) for all U ∈ 2 ^X and α ∈ L.

From Corollary 3.4 and (3) of Lemma 3.2 it is easy to prove the following theorem.

Theorem 3.5. Let (X, H₁) and (Y, H₂) be M-fuzzifying G-filter spaces. Then f^→ : (X, ε (H₁)) → (Y, ε (H₂)) is an LM-G-filter map if and only if f : (X, H₁) → (Y, H₂) is an M-fuzzifying G-filter map.

The following is an immediate consequence of Theorem 3.3 and Theorem 3.5:

Remark 3.6. ε is a functor from M-FYG to LM-G.

To a given LM-G-filter space, an M-fuzzifying G-filter space can be associated as shown below:

Theorem 3.7. Let G be an LM-G-filter on X. Then

ɛ (G) :2 ^X  → M defined by ɛ (G) (U) = ⋁ _{⊓i∈IU i =U} ⋀ _i∈I ⋁ {G (A) ; A ∈ L ^X , δ₁ (A) = U _i } for all U ∈ 2 ^X is an M-fuzzifying G-filter on X.

Proof. Clearly ɛ (H) (X) =1 . Let U, V ∈ 2 ^X such that U ⊆ V. If there exists A ∈ L ^X such that δ₁ (A) = U, define B ∈ L ^X by B (x) =1 if x ∈ V and B (x) = A (x) if x ∈ X \ V. Then δ₁ (B) = V and B ≥ A. Therefore, whenever U ≤ V, ⋁ {G (A) ; A ∈ L ^X , δ₁ (A) = U} ≤ ⋁ {G (B) ; B ∈ L ^X , δ₁ (B) = V} . Since U ≤ V, for every finite collection {U _i  ; ⊓ _i∈IU _i  = U}, we have the finite collection {V _i  = (U _i  ⋃ V) ; ⊓ _i∈IV _i  = V}. Hence ɛ (G) (U) ≤ ɛ (G) (V) . By (1) of Lemma 3.2 it is easy to prove that for U, V ∈ 2 ^X , ɛ (G) (U ∧ V) ≥ ɛ (G) (U) ∧ ɛ (G) (V). Hence ɛ (G) is an M-fuzzifying G-filter on X.

Second part of the proof is trivial.

By (3) of Lemma 3.2, it is easy to prove the following theorem:

Theorem 3.8. Let (X, G₁) and (Y, G₂) be LM-G-filter spaces. If f^→ : (X, G₁) → (Y, G₂) is an LM-G-filter map then both f : (X, ɛ (G₁)) → (Y, ɛ (G₂)) and f : (X, [G₁]) → (Y, [G₂]) are M-fuzzifying G-filter maps.

Remark 3.9. From Theorem 3.7 and Theorem 3.8, it is easy to observe that ɛ, [.] are functors from LM-G to M-FYG. Definition 3.10. An LM-G-filter space(X, G) is called catalyzed LM-G-filter space if G (A) = G (1_δ1(A)) for all A ∈ L ^X (i.e. G = ε ([G])). Example 3.11. Let X = {1, 2, 3, 4} ,  L = [0, 1] ,  M = {0, 1}. G is defined by G (A) =1 if 1_{2} ≤ A and 0 otherwise. If A ∈ L ^X is such that 1_{2} ≤ A, then 1_{2} ≤ 1_δ1(A). Therefore G (A) = G (1_δ1(A)) for all A ∈ L ^X . If G (A) =0, then 1_{2} ≰ A. This implies 1_{2} ≰ 1_δ1(A). Therefore G (1_δ1(A)) =0. Therefore G (A) = G (1_δ1(A)) for all A ∈ L ^X . Therefore G is catalyzed LM-G-filter on X. Let CLM-G denotes the category of catalyzed LM-G-filter spaces and CLM-G(X) denotes the set of all catalyzed LM-G-filters on X.

Remark 3.12. Given an M-fuzzifying G-filter space (X, H), the associated LM-G-filter space (X, ε (H)) defined in Theorem 3.3 is a catalyzed LM-G-filter space.

Remark 3.13. Let (X, G) be an LM-G-filter space. Since 1_δ1(A) ≤ 1_{δ p (A)} for all A ∈ L ^X ,  G (1_δ1(A)) ≤ ⋀ _p∈pr(L)G (1_{δ p (A)}). Therefore every catalyzed LM-G-filter space is a weakly inspired LM-G-filter space.

Remark 3.14. Every weakly inspired LM-G-filter space is not a catalyzed LM-G-filter space.

For example, let X = [0, 1] and L = M = [0, 1]. Let I ∈ L ^X be defined by I (x) = x for all x ∈ X. Define G : L ^X  → M by

G ( A ) = { 1 if A = [ 0 , 1 ] , 0.2 + ( 1 - p ) 2 if A = [ p , 1 ] where p is prime and p ≠ 0 , G ( C ) if A contains crisp sets of the form [ p , 1 ] and C is the largest such crisp set , 0.2 if A doesn ′ t contain crisp sets of the form [ p , 1 ] and A ≥ I , 0.1 otherwise

It is clear that G is an LM-G-filter and G (A) = ⋀ _p∈pr(L)G (1_{δ p (A)}) for all A ∈ L ^X . Therefore (X, G) is an inspired LM-G-filter space and hence weakly inspired. For I ∈ L ^X , δ₁ (I) = {1}. But G (I) =0.2 . and G (1_{1}) =0.1. Therefore (X, G) not catalyzed.

The functors ɛ and ε are related by the following theorem:

Theorem 3.15.

ɛ o ε = id_M-FYG .

Proof.

Let H be an M-fuzzifying G-filter on X. Since ⋁ {ε (H) (A) : A ∈ L ^X , δ₁ (A) = U} = H (U) and H is an M-fuzzifying G-filter,

( ɛ o ε ) ( H ) ( U ) = ⋁ ⊓ i ∈ I V i = U ⋀ i ∈ I ⋁ { ε ( H ) ( A ) : A ∈ L X , δ 1 ( A ) = V i } = ⋁ ⊓ i ∈ I V i = U ⋀ i ∈ I H ( V i ) = H ( U )

Hence (ɛ o ε) (H) = H .

Combining Corollary 3.16 and Theorem 2.19 we get the following:

Theorem 3.17. ILM-G is isomorphic to CLM-G.

Remark 3.18. Let G be an inspired LM-G-filter on X. Then the corresponding catalyzed LM-G-filter is given by (ε o η) (G). i.e. if G is an inspired LM-G-filter on X, then the corresponding catalyzed LM-G-filter is defined by (ε o η) (G) (A) = G (1_δ1(A)) for all A ∈ L ^X . Let G be a catalyzed LM-G-filter on X. Then the corresponding inspired LM-G-filter is given by (ζ o ɛ) (G). i.e. if G is a catalyzed LM-G-filter on X, then the corresponding inspired LM-G-filter is defined by (ζ o ɛ) (G) (A) = ⋀ _p∈pr(L)G (1_{δ p (A)}) for all A ∈ L ^X .

Remark 3.19. Theorem 3.17 doesn’t imply that every catalyzed LM-G-filter space is inspired or vice-versa. For example, let X = [0, 1] and L = M = [0, 1]. Define G : L ^X  → M by

G ( A ) = { 1 if A = [ 0 , 1 ] , 0.2 + ( 1 - p ) 2 if A = [ p , 1 ] where p is prime and p ≠ 0 , G ( C ) if A contains crisp sets of the form [ p , 1 ] and C is the largest such crisp set , 0.1 otherwise

It is clear that G is an LM-G-filter and G (A) = G (1_δ1(A)) for all A ∈ L ^X so that G is catalyzed. For I ∈ L ^X defined by I (x) = x for all x ∈ X, G (I) =0.1,  δ _p  (I) = [p, 1] for all prime p and hence ⋀_p∈pr(L)G (1_{δ p (I)}) =0.2. Therefore G is not inspired.

It has already been noticed in Remark 3.14 that every inspired LM-G-filter space is not catalyzed.

Theorem 3.20. ɛ : LM-G →M-FYG is the right adjoint left inverse functor of ε : M-FYG →LM-G.

Proof. Let (X, H) be an M-fuzzifying G-filter space and (Y, G) be an LM-G-filter space. Let f^→ : (X, ε (H)) → (Y, G) be an LM-G-filter map. Then by Theorem 3.8, f : (X, ɛ (ε (H))) → (Y, ɛ (G)) is an M-fuzzifying G-filter map. i.e. f : (X, H) → (Y, ɛ (G)) is an M-fuzzifying G-filter map.

If g : (X, H) → (Y, ɛ (G)) is an M-fuzzifying G-filter map, then by Theorem 3.5, g^→ : (X, ε (H)) → (Y, ε (ɛ (G))) is an LM-G-filter map. Therefore, by (ii.) of Theorem 3.5, g^→ : (X, ε (H)) → (Y, G) is an LM-G-filter map. Hence, by Theorem 3.5, ɛ is the right adjoint left inverse functor of ε .

Since ε : M-FYG →LM-G is a left adjoint functor, we have the following results.

Corollary 3.21. Let {(X, H _j )} _j∈J be a family of M-fuzzifying G-filter spaces. Then ε (⋁ _j∈JH _j ) = ⋁ _j∈Jε (H _j ) .

Corollary 3.22. Let {(X _j , H _j )} _j∈J be a family of M-fuzzifying G-filter spaces, different X j ′ s be disjoint. Then ε (oplus _j∈JH _j ) = oplus _j∈Jε (H _j ) .

Corollary 3.23. Let (X, H) be an M-fuzzifying G-filter space and (Y, H/f) be the M-fuzzifying quotient G-filter space of (X, H) with respect to the surjective mapping f : X → Y . Then ε (H/f) = ε (H)/f^→ .

Theorem 3.24. [.] : CLM-G→M-FYG is the left adjoint functor of ε : M-FYG→CLM-G.

Proof. Let (X, G) be a catalyzed LM-G-filter space and (Y, H) be an M-fuzzifying G-filter space. If f : (X, [G]) → (Y, H) is an M-fuzzifying G-filter map, then by Theorem 3.5 f^→ : (X, ε ([G])) → (Y, ε (H)) is an LM-G-filter map. Hence, for all B ∈ L ^Y , G (f^← (B)) = ε ([G]) (f^← (B)) ≥ ε (H) (B). Therefore, f^→ : (X, G) → (Y, ε (H)) is an LM-G-filter map.

If g^→ : (X, G) → (Y, ε (H)) is an LM-G-filter map, then [G] (g^-1 (U)) = G (1_g^-1(U)) = G (g^← (1 _U )) ≥ ε (H) (1 _U ) = H (U) for all U ∈ 2 ^X  . Therefore, g : (X, [G]) → (Y, H) is an M-fuzzifying G-filter map. Therefore, [.] : CLM-G→M-FYG is the left adjoint functor of ε : M-FYG→CLM-G.

Since ε : M-FYG →CLM-G is a right adjoint functor, we have the following results.

Corollary 3.25. Let {(X, H _j )} _j∈J be a family of M-fuzzifying G-filter spaces. Then ε (⋀ _j∈JH _j ) = ⋀ _j∈Jε (H _j ) .

Corollary 3.26. Let {(X _j , H _j )} _j∈J be a family of M-fuzzifying G-filter spaces and X = ∏ j ∈ J X j . Then ε ( ∏ j ∈ J H j ) = ∏ j ∈ J ε ( H j ) .

Corollary 3.27. Let (X, H) be M-fuzzifying G-filter space and Y ⊆ X . Then ε (H) | _Y  = ε (H| _Y ) .

Remark 3.28. From Theorem 3.5, Corollary 3.25 and Corollary 3.21 it is easy to prove that both ε : M-FYG(X) →CLM-G(X) and ɛ : CLM-G(X) →M-FYG(X) are complete lattice isomorphisms.

4 Catalyzation of weakly inspired LM-G-filter spaces

This section derives some basic properties of catalyzed LM-G-filter spaces and identifies CLM-G as an isomorphism-closed, bireflective and bicoreflective full subcategory of WILM-G.

Theorem 4.1. Let (X, G) be a catalyzed LM-G-filter space and Y ⊆ X. Then (Y, G| _Y ) is also catalyzed.

Proof. By Corollary 3.27, G| _Y  = ε ([G]) | _Y  = ε ([G] | _Y ) ≤ ε ([G| _Y ]) . Therefore (G| _Y ) (A) ≤ (G| _Y ) (1_δ1(A)) for all A ∈ L ^X . Since G| _Y is an LM-G-filter and 1_δ1(A) ≤ A,   (G| _Y ) (1_δ1(A)) ≤ (G| _Y ) (A).

Theorem 4.2. Let {(X _j , G _j )} _j∈J be a family of LM-G-filter spaces and X = ∏ j ∈ J X j . If (X _j , G _j ) is catalyzed for all j ∈ J, then ( X , ∏ j ∈ J G j ) is catalyzed.

Proof. By Corollary 3.27, ∏ j ∈ J G j = ∏ j ∈ J ε ( [ G j ] ) = ε ( ∏ j ∈ J [ G j ] ) ≤ ε ( [ ∏ j ∈ J G j ] ) . Reverse inequality is trivial.

Theorem 4.3. Let {(X _j , G _j )} _j∈J be a family of LM-G-filter spaces, different X j ′ s be disjoint. Then oplus _j∈J (X _j , G _j ) is catalyzed if and only if (X _j , G _j ) is catalyzed for all j ∈ J .

Proof. By Corollary 3.22, ε ([oplus _j∈JG _j ]) = ε (oplus _j∈J [G _j ]) = oplus _j∈Jε ([G _j ]). If oplus _j∈JG _j is catalyzed, then oplus _j∈JG _j  = ε ([oplus _j∈JG _j ]) = oplus _j∈Jε ([G _j ]) . Hence, G _j  = oplus _j∈JG _j | _{X j}  = oplus _j∈Jε ([G _j ]) | _{X j}  = ε ([G _j ]). Therefore, G _j is catalyzed. Conversely, if G _j is catalyzed for all j ∈ J, then ε ([oplus _j∈JG _j ]) = oplus _j∈Jε ([G _j ]) = oplus _j∈JG _j  .

The following theorems are immediate consequences of Corollaries, 3.23, 3.25 and 3.21 respectively.

Theorem 4.4. Let (X, G) be an LM-G-filter space and (Y, G/f^→) be the quotient LM-G-filter space of (X, G) with respect to the surjective mapping f : X → Y . If (X, G) is catalyzed, then (Y, G/f^→) is catalyzed.

Theorem 4.5. Let {(X, G _j )} _j∈J be a family of LM-G-filter spaces. If (X, G _j ) is catalyzed for all j ∈ J, then (X, ⋀ _j∈JG _j ) is also catalyzed.

Theorem 4.6. Let {(X, G _j )} _j∈J be a family of LM-G-filter spaces. If (X, G _j ) is catalyzed for all j ∈ J, then (X, ⋁ _j∈JG _j ) is also catalyzed.

Now we obtain a left adjoint functor σ and a right adjoint functor ς from WILM-G to CLM-G for the inclusion functor i : CLM-G→WILM-G and prove that CLM-G is an isomorphism-closed, bireflective and bicoreflective full subcategory of WILM-G.

Theorem 4.7. Let (X, G) be a weakly inspired LM-G-filter space and σ (G) : L ^X  → M be defined by σ (G) = ε ([G]) . Then σ (G) is the finest catalyzed LM-G-filter coarser than G and σ (G) is referred to as the interior catalyzation of G.

Proof. From the definition of σ (G) it is clear that σ (G) is an LM-G-filter on X coarser than G. σ (G) is catalyzed since ε ([σ (G)]) = ε ([ε ([G])]) = ε ([G]) = σ (G) . Let G ˆ be any catalyzed LM-G-filter coarser than G. Since G ˆ is catalyzed, we have G ˆ = ε ( [ G ˆ ] ) . Therefore G ˆ = ε ( [ G ˆ ] ) ≤ ε ( [ G ] ) = σ ( G ) .

Theorem 4.8. Let (X, G₁) and (X, G₂) be weakly inspired LM-G-filter spaces. If f^→ : (X, G₁) → (Y, G₂) is an LM-G-filter map, then f^→ : (X, σ (G₁)) → (Y, σ (G₂)) is an LM-G-filter map.

Proof. Since f^→ : (X, G₁) → (Y, G₂) is an LM-G-filter map, f : (X, [G₁]) → (Y, [G₂]) is an M-fuzzifying G-filter map. Then by Theorem 3.5 f^→ : (X, ε ([G₁])) → (Y, ε ([G₂])) is an LM-G-filter map. Therefore, σ (G₁) (f^← (B)) = ε ([G₁]) (f^← (B)) ≥ ε ([G₂] (B) = σ (G₂) (B) for all B ∈ L ^Y . Remark 4.9. From Theorem 4.7 and Theorem 4.8 it is obvious that σ is a functor from WILM-G to CLM-G.

Lemma 4.10. σ : WILM-G →CLM-G is the left adjoint functor of the inclusion functor i : CLM-G→WILM-G.

Proof. Let (X, G₁) be a weakly inspired LM-G-filter space and (Y, G₂) be a catalyzed LM-G-filter space. If f^→ : (X, σ (G₁)) → (Y, G₂) is an LM-G-filter map, then for B ∈ L ^Y , G₁ (f^← (B)) ≥ σ (G₁) (f^← (B)) ≥ G₂ (B). Hence f^→ : (X, G₁) → (Y, i (G₂)) is an LM-G-filter map.

If g^→ : (X, G₁) → (Y, i (G₂)) is an LM-G-filter map, then by Theorem 4.8 g^→ : (X, σ (G₁)) → (Y, G₂) is an LM-G-filter map. Hence, σ : WILM-G →CLM-G is the left adjoint functor of the inclusion functor i : CLM-G→WILM-G.

Theorem 4.11. CLM-G is a bireflective full subcategory of WILM-G.

Proof. Since for a weakly inspired LM-G-filter space (X, G) , id _X  : (X, G) → (X, σ (G)) is the CLM-G reflection and it is bijective, CLM-G is a bireflective full subcategory of LM-G.

Though the following results are obvious, we record them for future reference.

Theorem 4.12. CLM-G is an isomorphism-closed full subcategory of WILM-G.

Since σ : WILM-G →CLM-G is a left adjoint functor, we have the following results.

Corollary 4.13. Let {(X, G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces. Then σ (⋁ _j∈JG _j ) = ⋁ _j∈Jσ (G _j ) .

Corollary 4.14. Let {(X _j , G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces, different X j ′ s be disjoint. Then σ (oplus _j∈JG _j ) = oplus _j∈Jσ (G _j ) .

Corollary 4.15. Let (X, G) be a weakly inspired LM-G-filter space and (Y, G/f^→) be the quotient LM-G-filter space of (X, G) with respect to the surjective mapping f : X → Y . Then σ (G/f^→) = σ (G)/f^→ .

Theorem 4.16. Let {(X, G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces. Then σ (⋀ _j∈JG _j ) = ⋀ _j∈Jσ (G _j ) .

Theorem 4.17. Let (X, G) be a weakly inspired LM-G-filter space and Y ⊆ X . Then σ (G) | _Y  ≤ σ (G| _Y ) .

Theorem 4.18. Let {(X _j , G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces and X = ∏ j ∈ J X j . Then ∏ j ∈ J σ ( G j ) ≤ σ ( ∏ j ∈ J G j ) .

We leave the following question open.

Question 4.19. Does equality hold in Theorem 4.17 and Theorem 4.18 ?

Theorem 4.20. Let (X, G) be a weakly inspired LM-G-filter space and ψ ^G  : L ^X  → M be defined by ψ ^G  (A) = ⋁ {G (B) ; B ∈ L ^X ,  1_δ1(B) = 1_δ1(A)}. Then ς (G) (A) = ⋁ _{⊓i∈IA i =A} ⋀ _i∈I ψ ^G  (A _i ) is the coarsest catalyzed LM-G-filter on X finer than G and ς (G) is referred to as the exterior catalyzation of G.

Proof. From the definition of ψ ^G it is clear that for all A ∈ L ^X , ψ ^G  (A) = ψ ^G  (1_δ1(A)). ς (G) (1 _X ) ≥ ψ ^G  (1 _X ) =1. Let U and V be characteristic functions such that U ≤ V. Let A ∈ L ^X be such that 1_δ1(A) = U. Define B ∈ L ^X by B (x) = A (x) if x ∈ X \ V and B (x) =1 if x ∈ V. Clearly 1_δ1(B) = V and B ≥ A. Therefore, ψ ^G  (U) ≤ ψ ^G  (V) . Let A, B ∈ L ^X such that A ≤ B. Then ψ ^G  (A) = ψ ^G  (1_δ1(A)) ≤ ψ ^G  (1_δ1(B)) = ψ ^G  (B). Therefore, whenever A ≤ B, we have ψ ^G  (A) ≤ ψ ^G  (B) . Since ς (G) (A) = ⋁ _{⊓i∈IB i =A} ⋀ _i∈I ψ ^G  (B _i ), it is easy to prove that ς (G (A)) ≤ ς (G (B)) whenever A ≤ B. It is clear from the definition of ς (G) that for A, B ∈ L ^X , ς (G) (A ∧ B) ≥ ς (G) (A) ∧ ς (G) (B). Therefore, ς (G) is an LM-G-filter on X.

Since ψ ^G  (A) = ψ ^G  (δ₁ (A)) for all A ∈ L ^X , ς ( G ) ( A ) = ⋁ ⊓ i ∈ I A i = A ⋀ i ∈ I ψ G ( A i ) = ⋁ ⊓ i ∈ I A i = A ⋀ i ∈ I ψ G ( 1 δ 1 ( A i ) ) ≤ ⋁ ⊓ i ∈ I A i = A ⋀ i ∈ I ς ( G ) ( 1 δ 1 ( A i ) ) ≤ ς ( G ) ( 1 δ 1 ( A ) ) Since ς (G) is an LM-G-filter and 1 _{δ 1}  (A) ≤ A, ς (G) (1_δ1(A)) ≤ ς (G) (A). Therefore ς (G) (A) = ς (G) (1_δ1(A)) for all A ∈ L ^X . Therefore, ς (G) is catalyzed. Since ς (G) (A) ≥ ψ ^G  (A) ≥ G (A) for all A ∈ L ^X , ς (G) is a catalyzed LM-G-filter on X finer than G. Let G ˆ be any catalyzed LM-G-filter on X finer than G. Then for all A ∈ L ^X , ψ G ( A ) = ⋁ { G ( B ) : B ∈ L X , 1 δ 1 ( B ) = 1 δ 1 ( A ) } ≤ ⋁ { G ˆ ( B ) : B ∈ L X , 1 δ 1 ( B ) = 1 δ 1 ( A ) } = ⋁ { G ˆ ( 1 δ 1 ( B ) ) : B ∈ L X , 1 δ 1 ( B ) = 1 δ 1 ( A ) } = G ˆ ( 1 δ 1 ( A ) ) = G ˆ ( A ) . Thus for all A ∈ L X , ψ G ( A ) ≤ G ˆ ( A ) . Therefore ς ( G ) ( A ) ≤ G ˆ ( A ) .

Theorem 4.21. Let (X, G₁) and (Y, G₂) be weakly inspired LM-G-filter spaces. If f^→ : (X, G₁) → (Y, G₂) is an LM-G-filter map, then f^→ : (X, ς (G₁)) → (Y, ς (G₂)) is an LM-G-filter map.

Proof. Let f^→ : (X, G₁) → (Y, G₂) be an LM-G-filter map and B ∈ L ^Y . If B is a characteristic function then f^← (B) is also a characteristic function in L ^X . Therefore ψ G 2 ( B ) = ⋁ { G 2 ( C ) , C ∈ L Y ; 1 δ 1 ( C ) = B } ≤ ⋁ { G 1 ( f ← ( C ) ) ; 1 δ 1 ( f ← ( C ) ) = f ← ( B ) } ≤ ψ G 1 ( f ← ( B ) ) ≤ ς ( G 1 ( f ← ( B ) ) Therefore, for all B ∈ L ^Y , ψ ^{G ₂}  (B) = ψ ^{G ₂}  (1_δ1(B)) ≤ ς (G₁ (f^← (1_δ1(B))) = ς (G₁ (f^← (B)) .

The following remark is an immediate consequence of Theorem 4.20 and Theorem 4.21.

Remark 4.22. ς is a functor from WILM-G to CLM-G.

Lemma 4.23. ς : WILM-G →CLM-G is the right adjoint functor of the inclusion functor i : CLM-G→WILM-G.

Proof. Let (X, G₁) be a catalyzed LM-G-filter space and (Y, G₂) be a weakly inspired LM-G-filter space. If f^→ : (X, i (G₁)) → (Y, G₂) is an LM-G-filter map, then by Theorem 4.21, f^→ : (X, G₁) → (Y, ς (G₂)) is an LM-G-filter map.

If g^→ : (X, G₁) → (Y, ς (G₂)) is an LM-G-filter map, then G₁ (f^← (B)) ≥ ς (G₂) (B) ≥ G₂ (B) for all B ∈ L ^Y . Hence f^→ : (X, i (G₁)) → (Y, G₂) is an LM-G-filter map. Therefore, ς : WILM-G →CLM-G is the right adjoint functor of the inclusion functor i : CLM-G→WILM-G.

Theorem 4.24. CLM-G is a bicoreflective full subcategory of WILM-G.

Proof. Since for a weakly inspired LM-G-filter space (X, G), the identity mapping id _X  : (X, ς (G)) → (X, G) is the CLM-G coreflection and it is bijective, CLM-G is a bicoreflective full subcategory of WILM-G.

Since ς : WILM-G →CLM-G is a right adjoint functor, we have the following results.

Corollary 4.25. Let {(X _j , G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces. Then ς (⋀ _j∈JG _j ) = ⋀ _j∈Jς (G _j ).

Corollary 4.26. Let {(X _j , G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces and X = ∏ j ∈ J X j . Then ς ( ∏ j ∈ J G j ) = ∏ j ∈ J ς ( G j ) .

Corollary 4.27. Let (X, G) be a weakly inspired LM-G-filter space and Y ⊆ X . Then ς (G) | _Y  = ς (G| _Y ) .

Theorem 4.28. Let {(X _j , G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces, different X j ′ s be disjoint. Then ς (oplus _j∈JG _j ) = oplus _j∈Jς (G _j ).

Proof. From Theorem 4.3 and Theorem 4.20 it is clear that ς (oplus _j∈JG _j ) ≤ oplus _j∈Jς (G _j ). Conversely, let A ∈ L ^X and c∈ co-pr(L) be such that c ⪯ oplus _j∈Jς (G _j ) (A). Then c ⪯ oplus j ∈ J ς ( G j ) ( A ) = ⋀ j ∈ J ς ( G j ) ( A | X j ) = ⋀ j ∈ J ⋁ ⊓ i ∈ I j B ji = A | X j ⋀ i ∈ I j ψ j G ( B ji ) . Then for every j ∈ J, there exists {B _ij } _{i∈I j}  ⊆ L ^{X _j} such that ⊓_{i∈I j} B _ji  = A| _{X j} and for each such i ∈ I _j , c ≤ ψ j G ( B ji ) . Let B ij * ∈ L X be defined by B ij * ( x ) = { B ij x ∈ X j , 1 x ∉ X j Then ⊓ i ∈ I j B ji * = A and ψ G j ( B ji ) = ψ oplus j ∈ J G j ( B ji * ) . Therefore c ≤ ψ oplus j ∈ J G j ( B ji * ) for each i ∈ I _j . Therefore, c ≤ ⋀ i ∈ I j ψ oplus j ∈ J G j ( B ji * ) . Hence, c ≤ ⋁ _{⊓i∈I j C ji =A} ⋀ _{i∈I j} ψ^{oplus_j∈JG _j}  (C _ji ) = ς (oplus _j∈JG _j ) (A). Therefore, oplus _j∈Jς (G _j ) ≤ ς (oplus _j∈JG _j ).

The following theorems can also be easily proved.

Theorem 4.29. Let {(X _j , G _j )} _j∈J be a family of weakly inspired LM-G-filter spaces. Then ς (⋁ _j∈JG _j ) ≤ ⋁ _j∈Jς (G _j ).

Theorem 4.30. Let (X, G) be a weakly inspired LM-G-filter space and (Y, G/f^→) be the quotient LM-G-filter space of (X, G) with respect to the surjective mapping f : X → Y . Then ς (G/f^→) ≤ ς (G)/f^→.

We leave the following question also open.

Question 4.31. Does equality hold in Theorem 4.29 and Theorem 4.30 ?

Remark 4.32. Since the objects in CLM-G are structured sets, CLM-G is a construct. But CLM-G is not topological. Let X = {x} ,  L = M = [0, 1]. Then L ^X  = {α _X   ; α ∈ [0, 1]}. G₁ and G₂ defined by G 1 ( α X ) = { 1 if α = 1 , 0.2 if α ∈ [ 0 , 1 ) G 2 ( α X ) = { 1 if α = 1 , 0.3 if α ∈ [ 0 , 1 ) are catalyzed LM-G-filters on {x}. Hence terminal separator property doesn’t hold for CLM-G. Therefore CLM-G is not topological.

5 An application

Let X = {p₁, p₂, p₃} be the set of three pests which attack the set of four cereals Y = {c₁, c₂, c₃, c₄} and Z = {p₁, p₂}. Let L = ([0, 1] , ∧ , ∨ , 0, 1) be a completely distributive lattice with x ∧ y = min(x, y) and x ∨ y = max(x, y) and M be the lattice shown in Fig. 1.

OPEN IN VIEWER

Fig. 1

The diamond type lattice

Consider the fuzzy information F ∈ [0, 1] ^X×Y where F (p _i , c _j ) gives the extent to which the pest p _i destroys the cereal c _j as given in Table 1. Note that pests p₁ and p₂ are extremely harmful whereas p₃ is less harmful.

Let R : X × X → [0, 1] be a fuzzy relation where R (p _i , p _j ) represents the degree to which a cereal attacked by pest p _i is prone to the attack of pest p _j as given in Table 2.

{Let each A ∈ L ^Z denotes a pesticide such that A (p _i ) represents the degree to which pesticide A can destroy pest p _i . Our aim is to design a mathematical model for effective distribution of a given funding for the development of various pesticides giving high priority to pesticides which eliminate the harmful pests p₁ or p₂ completely (i.e. either A (p₁) =1 or A (p₂) =1). If pesticide B is more effective in destroying the pests p₁ and p₂ than pesticide A, then priority is given to the production of pesticide B naturally.∥The nonzero values for R (p₃, p _i ) and R (p _i , p₃) ,  i ∈ {1, 2} point out that a cereal attacked by the less harmful pest p₃ is prone to the attack of harmful pests p₁ and p₂ and vice versa. Thus, even if a particular pesticide is not effective in destroying neither p₁ nor p₂, its production matters if it can completely eliminate p₃ (i.e. the extension B : X → [0, 1] where B (p₃) =1 of the zero function 0 _Z  : Z → L, effectively destroys the pest p₃ though it fails to eliminate both p₁ and p₂).∥Suppose the financial assistance for the development of pesticide A _i , i ∈ {1, 2, 3, 4} is calculated using the map G as shown in Table 3 where A₁ (0 _Z ) ,  A₂,  A₃ and A₄ (1 _Z ) are the characteristic functions in L ^Z . Now G can be extended to all other A ∈ L ^Z by G (A) = G (1_δ1(A)). It is clear that G : L ^Z  → M is a catalyzed LM-G-filter which satisfies all the requirements for effective distribution of a given funding for the development of various pesticides. Thus, each catalyzed LM-G-filter models appropriate distribution of the available funding for the selective production of various pesticides.

The concept of LM-G-filter spaces is a generalization of LM-filter spaces. The study has introduced catalyzed LM-G-filter spaces, a subcategory of LM-G-filter spaces, and studied the properties of these spaces. This study identifies categorical connections of catalyzed LM-G-filter spaces with ILM-G (the category of inspired LM-G-filter spaces) and WILM-G (the category of weakly inspired LM-G-filter spaces). It is also proved that CLM-G is isomorphic to ILM-G. Based on interior and exterior catalyzation of weakly inspired LM-G-filter spaces, the study finds out CLM-G as an isomorphism-closed, bireflective and bicoreflective full subcategory of WILM-G.

To put it briefly, the study establishes the categorical connections among various subcategories of LM-G as shown in Fig. 2, where r, c represent bireflective and bicoreflective full subcategories respectively and the parallel lines represent isomorphic categories.

OPEN IN VIEWER

Fig. 2

Relationship among various categories

Investigation into the categorical connections between LM-G and LM-FTop - the category of LM-fuzzy topological spaces - and checking, whether the identified subcategories of LM-G induce any subcategory of LM-FTop, are part of our future study.