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On the Categories of Weak and Strong LM-G-Filter Spaces

On the Categories of Weak and Strong LM-G-Filter Spaces FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 228–242 https://doi.org/10.1080/16168658.2022.2057065 a b Merin Jose and Sunil C. Mathew a b Department of Mathematics, St. Thomas College Palai, Kottayam, Kerala, India; Department of Mathematics, Deva Matha College Kuravilangad, Kottayam, Kerala, India ABSTRACT ARTICLE HISTORY Received 2 June 2021 In this paper, the authors introduce the notion of weak r-level LM-G- Accepted 20 March 2022 filter spaces and strong p-level LM-G-filter spaces and discuss certain properties of these spaces. The study identifies WLM -G, the category KEYWORDS of weak r-level LM-G-filter spaces as an isomorphism-closed bireflec- Weak r-level LM-G-filter tive full subcategory of LM-G, the category of LM-G-filter spaces. It is spaces; strong p-level also proved that SLM -G, the category of strong p-level LM-G-filter p LM-G-filter spaces; initial spaces is an isomorphism-closed bicoreflective full subcategory of structures; fibre smallness; reflective subcategory; LM-G. Moreover, level decompositions of LM-G-filter spaces are stud- coreflective subcategory ied and some properties of the associated L-pre G-filter spaces are obtained. 2020 MATHEMATICS SUBJECT CLASSIFICATIONS 54A20; 18A40; 06D22 1. Introduction The study of filter and topology is highly interrelated. Because of the great importance of filter theory in topology, the concept of filters has been extended to different types of fuzzy filters. In 1977, Lowen [1] introduced the concept of filters in I and called them prefilters. Later in 1999, Burton et al. [2] introduced the concept of generalised filters as a map from 2 to I. Subsequently, Höhle and Šostak [3] developed the notion of L-filters and stratified L-filters on a complete quasi-monoidal lattice. In 2006, Kim et al. [4] introduced the notion of L-filter based on a strictly two-sided, commutative quantale lattice L and identified two types of images and preimages of L-filter bases. Later, in 2013 Jäger [5] developed the the- ory of stratified LM-filters which generalises the theory of stratified L-filters by introducing stratification mapping, where L and M are frames. In [6], Abbas et al. investigated strati- fied L-filter structure where L is a strictly two-sided, commutative quantale lattice. In [7], the authors disproved certain known theorems on L-filters in [6] and the rectification of the same led to the introduction of LM-G-filter spaces where those results hold good. More- over, an LM-filter is constructed from a given LM-G-filter and a categorical relationship is established between them in [7]. Recent studies on LM-G-filter spaces can be found in [8,9]. In this paper, we introduce the concept of weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces. It is proved that WLM -G, the category of weak r-level LM-G- filter spaces is an isomorphism-closed bireflective full subcategory of LM-G, the category CONTACT Sunil C. Mathew sunilcmathew@gmail.com © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 229 of LM-G-filter spaces. It is also proved that SLM -G, the category of strong p-level LM-G- filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G.Wehavealso constructed L-pre G-filter spaces by level decompositions of LM-G-filter spaces and studied their properties. 2. Preliminaries Throughout this paper, X stands for a non empty set, L and M stand for completely distribu- tive 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 the various notions of category theory, the readers can refer to [10,11]. Definition 2.1 ([12]): 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 ([12]): 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 [12] 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. Theorem 2.3 ([12]): Let L be a lattice, an order reversing involution on L, then (1) co-pr(L) = pr(L) (2) pr(L) =co-pr(L) Definition 2.4 ([7]): An LM-G-filter on a set X is defined to be a mapping G : L → M satisfying, G1: G(1 ) = 1; G2: For every A, B ∈ L such that A ≤ B, G(A) ≤ G(B); G3: For every A, B ∈ L , G(A ∧ B) ≥ G(A) ∧ G(B). The pair (X, G) is called an LM-G-filter space. When M ={0, 1}, it is called an L-pre G-filter space and when L ={0, 1},itiscalledan M-fuzzifying G-filter space. If G and G are two LM-G-filters on X such that G (A) ≥ G (A) for all A ∈ L , then we say 1 2 2 1 (X, G ) is weaker than (X, G ) and (X, G ) is stronger than (X, G ). 1 2 2 1 Remark 2.5: In addition to the above axioms, if G4: G(0 ) = 0isalsosatisfied, then (X, G) becomes an LM-filter space. 230 M. JOSE AND S. C. MATHEW → X Y Definition 2.6 ([7]): Let (X, G ) and (Y, G ) be LM-G-filter spaces. A map f : L → L is 1 2 ← Y called an LM-G-filter map if G (f (B)) ≥ G (B) for all B ∈ L . 1 2 → X Y Definition 2.7 ([7]): Let (X, G ) and (Y, G ) be LM-G-filter spaces. A map f : L → L is 1 2 → X called an LM-G-filter preserving map if G (f (A)) ≥ G (A) for all A ∈ L . 2 1 L-pre G-filter maps and L-pre G-filter preserving maps in L-pre G-filter spaces are defined analogously. Notation 2.8 ([7]): Let LM-G denotes the category of LM-G-filter spaces where morphisms are LM-G-filter maps and LM-G(X) denotes the lattice of LM-G-filters on X. Definition 2.9 ([7]): Let (X, G) be an LM-G-filter space and Y ⊆ X. Then the LM-G-filter, G| X Y defined on Y by (G| )(B) = {G(A)|A ∈ L , A| = B} for all B ∈ L is called the subspace of Y Y (X, G). Notation 2.10: Throughout this paper,  stands for finite intersection. Definition 2.11 ([7]): Let {(X , G )} be a family of LM-G-filter spaces, X = X and j j j∈J j j∈J p : X → X be the projection map. Then the product of {(X , G )} is defined as j j j j j j∈J j∈J X ← X G(A) = {G (D); D ∈ L , p (D) = B } for all A ∈ L . The product space j i B ≤A i∈I j∈J j i∈I i is denoted by (X, G ). j∈J Definition 2.12 ([7]): Let (X, G) be an LM-G-filter space and f : X → Y be a surjective map- → → ← Y ping. Then the LM-G-filter, G/f defined on Y by G/f (B) = G(f (B)) for all B ∈ L is called quotient LM-G-filter of G with respect to f. Definition 2.13 ([7]): Let {(X , G )} be a family of LM-G-filter spaces, X s be pairwise j j j∈J disjoint and X = X . Then the LM-G-filter, G defined on X by G (A) = j j j j∈J j∈J j∈J G (A| ) for all A ∈ L is called sum LM-G-filter of {G } . j X j j∈J j∈J j Definition 2.14 ([11]): A category C is said to be topological if the following conditions are satisfied: (i) Existence of initial structures:Foranyset X,anyfamily ((X , ξ )) of C-objects indexed i i i∈I by a class I and any family (f : X → X ) of maps indexed by I there exists a unique i i i∈I C-structure ξ on X which is initial with respect to (f : X → X ) in the sense that for i i i∈I any C-object (Y, η),amap g : (Y, η) → (X, ξ) is a C-morphism iff for every i ∈ I the composite map f og : (Y, η) → (X , ξ ) is a C-morphism. i i i (ii) Fibre-smallness: For any set X,the class {ξ |(X, ξ) is a C-object} of all C-structures with underlying set X, C-fibre of X,isaset. (iii) Terminal separator property: For any set X with cardinality atmost one, there exists exactly one C-structure on X. FUZZY INFORMATION AND ENGINEERING 231 3. Level Decompositions of LM-G-Filter Spaces In this section, we study level decompositions of LM-G-filter spaces with respect to ≥ relation and relation and study the properties of associated L-pre G-filter spaces. It is easy to prove the following theorem which associates a family of L-pre G-filter spaces to a given LM-G-filter space. Theorem 3.1: Let (X, G) be an LM-G-filter space. Then for each α ∈ M, G ={A ∈ L ; G(A) ≥ (α) α} is an L-pre G-filter on X. Remark 3.2: It is also easy to verify that if (X, G) is an LM-filter space, then G is an L-pre (α) filter for all α> 0. Remark 3.3: The L-pre G-filter G is called α-level decomposition of the LM-G-filter G with (α) respect to ≥ relation. Let L-PG denotes {G ; α ∈ M} (i.e. the set of all associated L-pre G- (G) (α) filters of an LM-G-filter space (X, G) with respect to ≥ relation) and L-PF denotes {F ; α ∈ (F) (α) M} (i.e. the set of all associated L-pre filters of an LM-filter space (X, F) with respect to ≥ relation). Theorem 3.4: Let G be an LM-G-filter on X. Then L-PG is a complete lattice. (G) Proof: Let {G ; α ∈ M ⊆ M} be an arbitrary collection in L-PG . Then it is easy to (α) 1 (G) observe that G = G .Also L = G ≥ G for every α ∈ M. Therefore (α) ( α) (0) (α) α∈M α∈M L-PG is a complete lattice. (G) Remark 3.5: Proceeding in the same way, it is easy to prove that given an LM-filter space (X, F), L-PF is a complete meet semilattice. (G) The following theorem gives an expression for join of an arbitrary collection of L-pre G-filters in L-PG . (G) Theorem 3.6: Let (X, G) be an LM-filter space and {G ; α ∈ M ⊆ M} be an arbitrary col- (α) 1 lection of L-pre G-filters in L-PG . Then G = G where K ={k ∈ M; G ⊇ (G) (α) ( k) (k) α∈M k∈K G ∀ α ∈ M }. (α) 1 Proof: Since L-PG is a complete lattice and G ⊇ G for every k ∈ K, (G) (k) (α) α∈M G = G .Itisclear that G ⊆ G . The reverse inequality is (α) (k) ( k) (k) α∈M k∈K k∈K k∈K obvious since, for every A ∈ L , A ∈ G ⇒ G(A) ≥ k for every k ∈ K. Therefore A ∈ (k) k∈K G . Hence G = G . Therefore G = G . ( k) ( k) (k) (α) ( k) k∈K α∈M k∈K k∈K k∈K The following theorem relates an L-pre G-filter, G in L-PG with the collection of all (α) (G) G where β ∈ β (α). (β) Theorem 3.7: Let (X, G) be an LM-G-filter space. Then (i) G ⊆ G whenever β  α. (α) (β) (ii) G = G where β ∈co-pr(M). (α) (β) βα 232 M. JOSE AND S. C. MATHEW Proof: Proof of (i) is trivial. Let A ∈ G (β) βα ⇒ G(A) ≥ β ∀β  α ⇒ G(A) ≥ β = α βα ⇒ A ∈ G (α) Therefore, G = G where β ∈ co-pr(M). (α) (β) βα Given a descending family of L-pre G-filters on X, we can construct an LM-G-filter on X as follows: Theorem 3.8: Let {G ; α ∈co-pr (M)} be a family of L-pre G-filters on X such that G ⊇ (α) (α ) G whenever α  α . Then G : L → M defined by G(A) = {α ∈ co-pr(M); A ∈ G } is an (α ) 1 2 (α) LM-G-filter on X. Moreover if G = G , then G = G . (α) (β) (α) (α) βα Proof: Since 1 ∈ G for all α, G(1 ) = 1. Let A, B ∈ L such that A ≤ B. Therefore, if X (α) X A ∈ G for some α, then B ∈ G . Hence G(A) ≤ G(B). Now let A, B ∈ L such that (α) (α) G(A) = p and G(B) = q.Let N ={α ∈ co-pr(M); A ∈ G } and N ={α ∈ co-pr(M); B ∈ 1 (α) 2 G }. Then p ∧ q = [ α] ∧ [ β] = (α ∧ β) = l where each (α) α∈N β∈N α∈N ,β∈N lα∧β 1 2 1 2 l  α ∧ β for some α ∈ N and β ∈ N . For any l  α ∧ β, l  α and l  β. Therefore A, B ∈ 1 2 G . Hence A ∧ B ∈ G for all l. Therefore G(A ∧ B) ≥ G(A) ∧ G(B). Hence G is an LM-G-filter (l) (l) on X. Let G = G . It is clear that G ⊆ G . The reverse inequality is obtained as (α) (β) (α) (α) βα A ∈ G ⇒ G(A) ≥ α = β where β ∈ co-pr(M) (α) βα ⇒ β  α ≤ G(A) = {k ∈ co-pr(M); A ∈ G }∀β  α (k) ⇒∃k ∈ co-pr(M); A ∈ G such that k ≥ β ⇒ G ⊆ G (β) (k) ⇒ A ∈ G ∀β  α (β) ⇒ A ∈ G = G (β) (α) βα Corollary 3.9: Let G and H be two LM-G-filters on X. Then G = H if and only if G = H for (α) (α) all α ∈ co-pr(M). Remark 3.10: Theorems 3.7 and 3.8 are valid in the case of LM-filters and L-prefilters also. Given an LM-G-filter space (X, G), we can associate a family of L-pre G-filters with respect to as follows: FUZZY INFORMATION AND ENGINEERING 233 Figure 1. The diamond type lattice. (α) X Theorem 3.11: Let (X, G) be an LM-G-filter space. Then for each α ∈ M, G ={A ∈ L ; α G(A)} is an L-pre G-filter on X. (α) (α) X Proof: G(1 ) = 1and α  1. Therefore 1 ∈ G for each α ∈ M.Let A ∈ G and B ∈ L be X X (α) such that A ≤ B. Therefore G(A) ≤ G(B) and α  G(A). This implies α  G(B). Hence B ∈ G . (α) Let A, B ∈ G . This implies α  G(A) and α  G(B). This implies there exists S ⊆ M such that S ≤ G(A) but for all s ∈ S , s  α andthereexists S ⊆ M such that S ≤ G(B) but 1 1 1 1 2 2 for all s ∈ S , s  α. Therefore, S = S S with S ≤ G(A) G(B) ≤ G(A ∧ B), but for 2 2 2 1 2 (α) (α) all s ∈ S, s  α. Therefore A ∧ B ∈ G . Hence G is an L-pre G-filter on X. (α) X Remark 3.12: Let (X, G) be an LM-filter space. Then for each α ∈ M, G ={A ∈ L ; α G(A)} need not be an L-prefilter on X. Let L = [0, 1], M be the product lattice [0, 1] , X ={x, y} and G be an LM-filter on X. Then (α) (α) for α = (0, 0.2) ∈ M, (0, 0.2)  0 = G(0 ) which implies 0 ∈ G . Hence G is not a pre X X L-filter on X. (α) Remark 3.13: The L-pre G-filter G is called α-level decomposition of the LM-G-filter G (G) (α) with respect to relation. Let L-PG denotes {G ; α ∈ M} (i.e. the set of all associated L-pre G-filters of an LM-G-filter space (X, G) with respect to relation). (G) Remark 3.14: Given an LM-G-filter space (X, G), L-PG is not a lattice. Let X ={x} and X X L = M be the lattice shown in Figure 1. Then L ={0 , α , β , γ ,1 }. G : L → M defined X X X X X (0) (α) by G(α ) = α for all α ∈ L is an LM-G-filter on X. Then G ={α , β , γ ,1 }, G = X X X X X (β) (γ ) (1) (α) (β) {β , γ ,1 }, G ={α γ ,1 }, G ={1 }, G ={1 }. Therefore G ∧ G ={γ ,1 } X X X X X X X X X X (α) (G) whichcannotbeexpressedas G for any α ∈ M. Hence L-PG is not a lattice. (α) (G) The following theorem relates a L-pre G-filter, G in L-PG with the collection of all (β) G where β ∈ pr(L) such that β ≺ α. Theorem 3.15: Let (X, G) be an LM-G-filter space. Then, (α) (β) (i) G ⊆ G whenever β ≺ α. (α) (β) (ii) G = G where β ∈ pr(L). β≺α 234 M. JOSE AND S. C. MATHEW Proof: Proof of (i) is trivial. (α) (β) (β) It is clear that G ⊆ G .Let A ∈ G where β is prime. This implies β β≺α β≺α G(A) for all β ≺ α.If α G(A), then there exists p ∈ pr(L) such that α p G(A) which is (α) a contradiction. Therefore A ∈ G . Given a descending family of L-pre G-filters on X, we can construct an LM-G-filter on X as follows: (α) (α ) (α ) 1 2 Theorem 3.16: Let {G ; α ∈ pr(M)} be a family of L-pre G-filters on X such that G ⊇ G X (α) whenever α ≺ α . Then G : L → M defined by G(A) = {α ∈ pr(M); A ∈ G } is an LM-G- 1 2 (α) (β) (α) (α) filter on X. Moreover if G = G , then G = G . β≺α (α) X Proof: Since 1 ∈ G for all α ∈ pr(M), G(1 ) = 1. Let A, B ∈ L such that A ≤ B.So B ∈ X X (α) (α) G for some α implies A ∈ G . Hence G(A) = (α) α ≤ (α) α = G(B). Now let A∈G B∈G X (α) A, B ∈ L such that G(A) = p and G(B) = q.Let N ={α ∈ pr(M); A ∈ G } and N ={α ∈ 1 2 (α) pr(M); B ∈ G }. Then G(A) ∧ G(B) = p ∧ q = [ (α ) α ] ∧ [ (β ) β ] = (β ) (α ∧ β ) = i j (α ) i j i j j A∈G i B∈G A∈G ,B∈G (l) l.Since α ∧ β ≤ α , β ,and l α ∧ β , l canbesuchthat A, B ∈ G or i j i j i j l α ∧β ,l∈pr(M) i j (l) (l) (l) (l) (l) A ∈ G , B ∈ G or A ∈ G , B ∈ G or A, B ∈ G . This implies G(A) ∧ G(B) = l ≤ l α ∧β i j p. Therefore G(A ∧ B) ≥ G(A) ∧ G(B). Hence G is an LM-G-filter on X. (p) A∧B∈G (α) (β) (α) Let G = G .Let A ∈ G . This implies β≺α (α) (β) (β) (q) α  G(A).Since G = G , A ∈ G for all β ≺ α. Therefore A ∈ G implies α  q β≺α (α) and G(A) = q. This implies α  G(A). Therefore A ∈ G . The reverse inequality is α q obtained as, (α) Let A ∈ G . This implies α  G(A) = (α ) α . This implies α  α for every α such i i i A∈G (α ) (α ) (α ) i i i that A ∈ G . Therefore, whenever A ∈ G then α  α .Thusif α α then A ∈ G .This i i (α ) (α) shows that A ∈ G = G . α ≺α (α) (α) Corollary 3.17: Let G and H be two LM-G-filters on X. Then G = H if and only if G = H for all α ∈ pr(M). [β] X Theorem 3.18: Let (X, G) be an LM-G-filter space. Then for each β ∈ M, G ={A ∈ L ; β G (A)} is an L-pre G-filter on X where G (A) = (G(A)) and is the order reversing involution in M. Proof: For a, b ∈ M,itiseasytoprove that a  b if and only if a b . This implies a [β] X  X  (β) b if and only if a  b . Therefore G ={A ∈ L ; β  G (A)}={A ∈ L ; β  G(A)}= G which is an L-pre G-filter on X. Theorem 3.19: Let (X, G) be an LM-G-filter space. Then; [α] [β] (i) G ⊆ G whenever α  β. [α] [β] (ii) G = G where β ∈ co-pr(M). βα Proof: Proof follows from Theorems 2.3 and 3.15.  FUZZY INFORMATION AND ENGINEERING 235 It is also easy to observe that, [α] [α ] Theorem 3.20: Let {G ; α ∈ co-pr(M)} be a family of L-pre G-filters on X such that G ⊇ [α ] X  [β] G whenever α  α . Then G : L → M defined by G (A) = {β ∈ copr(M); A ∈ G } is an 1 2 [α] [β] [α] [α] LM-G-filter on X. Moreover if G = G , then G = G . βα 4. Weak r-level LM-G-Filter Spaces In this section we introduce the concept of weak r-level LM-G-filter spaces and iden- tify WLM -G(the category of weak r-level LM-G-filter spaces) as an isomorphism-closed bireflective full subcategory of LM-G. Definition 4.1: An LM-G-filter space (X, G) which takes only the values 0 or r, where r ∈ M for any 1 = A ∈ L is called weak r-level LM-G-filter space. Let WLM -G denotes the category of weak r-level LM-G-filter spaces where objects are weak r-level LM-G-filter spaces and morphisms are LM-G-filter maps. Theorem 4.2: WLM -G is an isomorphism-closed full subcategory of LM-G for each r ∈ M. Proof: Let (X, G ) be a weak r-level LM-G-filter space, (Y, G ) be an LM-G-filter space and 1 2 → → Y f : (X, G ) → (Y, G ) be an isomorphism. Since f is an isomorphism, for any B ∈ L such 1 2 ← ← ← that B = 1 , f (B) = 1 .If G (f (B)) = 0, then it is clear that G (B) = 0. If G (f (B)) = r, Y X 1 2 1 then G (B) = r since f is an isomorphism. Therefore (Y, G ) is a weak r-level LM-G-filter 2 2 space. We give left adjoint of the inclusion functor i : WLM -G → LM-G and show that WLM -G r r is a bireflective full subcategory of LM-G. r X Lemma 4.3: Let (X, G) be an LM-G-filter space. Then G : L → M defined by 1 if A = 1 , ⎨ X G (A) = rifG(A) ≥ r and A = 1 , 0 otherwise is the largest weak r-level LM-G-filter smaller than G. Remark 4.4: If G is a weak r-level LM-G-filter, then G = G . Lemma 4.5: Let (X, G) be an LM-G-filter space and (Y, H) be a weak r-level LM-G-filter space. → → r Then f : (X, G) → (Y, H) is an LM-G-filter map if and only if f : (X, G ) → (Y, H) is an LM-G- filter map. → Y Proof: Let f : (X, G) → (Y, H) be an LM-G-filter map. Let B ∈ L be such that H(B) = r. ← r ← Then G(f (B)) ≥ H(B) = r which implies G (f (B)) ≥ r = H(B). → r Y Conversely, let f : (X, G ) → (Y, H) be an LM-G-filter map. Let B ∈ L be such that r ← ← H(B) = r. Then G (f (B)) ≥ r which implies G(f (B)) ≥ r. Hence the proof.  236 M. JOSE AND S. C. MATHEW → r Let (X, G) and (Y, H) be LM-G-filter spaces. Then it is easy to prove that f : (X, G ) → r → r (Y, H ) is an LM-G-filter map if f : (X, G) → (Y, H) is an LM-G-filter map. Thus () is a functor from LM-G to WLM -G. Moreover it is easy to verify the following theorem. Theorem 4.6: () : LM-G → WLM -G is the left adjoint functor of i : WLM -G → LM-G. r r Theorem 4.7: WLM -G is a bireflective full subcategory of LM-G for all r ∈ M. Proof: Since for a LM-G-filter space (X, G), id : (X, G) → (X, G ) is the WLM -G reflection X r and it is bijective, WLM -G is a bireflective full subcategory of LM-G. Remark 4.8: The name weak r-level LM-G-filter space is justified since for an LM-G-filter r r space (X, G), id : (X, G) → (X, G ) is the WLM -G reflection and (X, G ) is an LM-G-filter X r space which is weaker than (X, G) and takes only values 0 or r for any 1 = A ∈ L . As every left adjoint functor preserves colimits, we have the following corollaries: r r Corollary 4.9: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J Corollary 4.10: Let {(X , G )} be a family of LM-G-filter spaces, different X s be disjoint. Then j j j∈J r r ( G ) = G . j j j∈J j∈J Corollary 4.11: Let (X, G) be an LM-G-filter space and (Y, G/f ) be the LM-quotient G-filter → X Y → r r → space of (X, G) with respect to the surjective mapping f : L → L . Then (G/f ) = G /f . r r Theorem 4.12: Let (X, G) be LM-G-filter space and Y ⊆ X. Then G | = (G| ) . Y Y r r Y r X Proof: It is clear that G | ≤ (G| ) .For B ∈ L ,let (G| ) (B) = r.Let I ={A ∈ L ; A | = Y Y Y i i Y B}. Then G(A ) ≥ r.Since A | = B for all i ∈ I, ( A )| = B.Since G( A ) ≥ i i Y i Y i i∈I i∈I i∈I r r G(A ), G ( A ) = r. Hence G | (B) = r. i i Y i∈I i∈I It is easy to prove the following theorems. r r Theorem 4.13: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J Theorem 4.14: Let {(X , G )} be a family of LM-G-filter spaces and X = X . Then j j j∈J j j∈J r r G ≤ ( G ) . j j j∈J j∈J Remark 4.15: The above inequality in Theorem 4.14 cannot be replaced by equality. For example, let X ={x , x }, Y ={y , y } and L be the diamond lattice shown in Figure 2 and 1 2 1 2 M be the lattice shown in Figure 1. X × Y ={(x , y ), (x , y ), (x , y ), (x , y )}. 1 1 1 2 2 1 2 2 FUZZY INFORMATION AND ENGINEERING 237 Figure 2. The lattice M . Let the LM-G-filter on X, G be defined by A A A A A A A A A A A A A A A A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x 0000 αααα β β β β 1111 x 0 αβ 10 αβ 10 αβ 10 αβ 1 G (A ) 0000 αααα β β β β γ γ γ 1 1 i Let the LM-G-filter on Y, G be defined by B B B B B B B B B B B B B B B B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 y 0000 αααα β β β β 1111 y 0 αβ 10 αβ 10 αβ 10 αβ 1 G (B ) 0000 ββββ α α α α γ γ γ 1 2 i → → X×Y X Y Let p and p be the projection maps from L to L and L respectively. Since 1 2 G (0 ) = γ , ( G ) (0 ) = γ . But G (0 ) = 0. j X j X j X j∈{1,2} j∈{1,2} j∈{1,2} Since the objects in WLM -G are structured sets, WLM -G is a construct and it has certain r r categorical properties as proved in the following theorem. Theorem 4.16: WLM -G satisfies the following properties: (i) Existence of initial structures (ii) Fibre-smallness. Proof: (i) Existence of initial structures :Let X be a nonempty set, {(X , G )} a family of j j j∈J weak r-level LM-G-filter spaces and {f : X → (X , G )} a family of maps. Then it is easy j j j j∈J to verify that G : L → M defined by G(A) = {G (P ); f (P ) = B } j j j j i B ≤A i∈I j∈J i∈I i is a weak r-level LM-G-filter on X such that each f : (X, G) → (X , G ) is an LM-G-filter j j j map. Let (Y, H) be a weak r-level LM-G-filter space. If g : (Y, H) → (X, G) is an LM-G- filter map, then it is obvious that the composite map (f og) : (Y, H) → (X , G ) is an j j j LM-G-filter map for each j ∈ J.If (f og) : (Y, H) → (X , G ) are LM-G-filter maps for each j j j 238 M. JOSE AND S. C. MATHEW j ∈ J then for each A ∈ L , G(A) = {G (P ); f (P ) = B } j j j j i B ≤A i∈I j∈J i∈I i ← ← ← ← ≤ {H((f og) (P )); g (f (P )) = g (B )} j j j j i B ≤A i∈I j∈J i∈I i ≤ H(g (A)) Therefore g : (Y, H) → (X, G) is an LM-G-filter map. (ii) Fibre-smallness : From Corollary 4.9 and Theorem 4.13, it is clear that for any nonempty set X,thesetofallweak r-level LM-G-filter spaces with underlying set X is a complete lattice with greatest element G defined by G(1 ) = 1and G(A) = r for all other A ∈ L . Therefore WLM -G is fibre small. Remark 4.17: WLM -G is not topological. Let X ={x}, L = M = [0, 1]. Then L ={α ; α ∈ r X [0, 1]}. G and G defined by 1 2 ⎧ ⎧ 1if α = 1 , 1if α = 1 , ⎨ X X ⎨ X X G (α ) = 0.4 if α ∈ [0.8, 1), G (α ) = 0.4 if α ∈ [0.7, 1), 1 X X 2 X X ⎩ ⎩ 0if α ∈ [0, 0.8) 0if α ∈ [0, 0.7) X X are weak 0.4-level LM-G-filters on X. Hence terminal separator property doesn’t hold for WLM -G. Therefore WLM -G is not topological. r r 5. Strong p-level LM-G-Filter Spaces In this section, we introduce the concept of strong p-level LM-G-filter spaces and iden- tify SLM -G (the category of strong p-level LM-G-filter spaces) as an isomorphism-closed bicoreflective full subcategory of LM-G. Definition 5.1: An LM-G-filter space (X, G) which takes only the values 1 or p, where p ∈ pr(M) for any 1 = A ∈ L is called strong p-level LM-G-filter space. Let SLM -G denotes the category of strong p-level LM-G-filter spaces where objects are strong p-level LM-G-filter spaces and morphisms are LM-G-filter maps. Theorem 5.2: SLM -G is an isomorphism-closed full subcategory of LM-G for each p ∈ pr(M). Proof: Let (X, G ) be an strong p-level LM-G-filter space, (Y, G ) be an LM-G-filter space 1 2 → → and f : (X, G ) → (Y, G ) be an isomorphism. Since f is an isomorphism, for any B ∈ 1 2 Y → ← ← ← L , G (B) = G (f (f (B))) ≥ G (f (B). Therefore, if G (f (B)) = 1, then G (B) = 1. If 2 2 1 1 2 ← ← G (f (B)) = p, then G (B) ≤ G (f (B)) = p. Therefore (Y, G ) is an strong p-level LM-G- 1 2 1 2 filter space. We give right adjoint of the inclusion functor i : SLM -G → LM-G and show that SLM -G p p is a bicoreflective full subcategory of LM-G. FUZZY INFORMATION AND ENGINEERING 239 p X Lemma 5.3: Let (X, G) be an LM-G-filter space. Then G : L → M be defined by pifG(A) ≤ p, G (A) = 1 if G(A)  p is the smallest strong p-level LM-G-filter larger than G. p X p Proof: Clearly G (1 ) = 1. Let A, B ∈ L such that A ≤ B.If G (A) = 1, then G(A)  p which p p p X implies G(B)  p and hence G (B) = 1. Therefore G (A) ≤ G (B).If A, B ∈ L such that p p G (A) = G (B) = 1, then G(A)  p and G(B)  p.Since p is prime, G(A) G(B)  p. There- fore G(A ∧ B)  p and hence G (A ∧ B) = 1. Remaining part of the proof is trivial. Remark 5.4: If G is a strong p-level LM-G-filter, then G = G . Lemma 5.5: Let (X, H) be a strong p-level LM-G-filter space and (Y, G) be an LM-G-filter space. → → p Then f : (X, H) → (Y, G) is an LM-G-filter map if and only if f : (X, H) → (Y, G ) is an LM- G-filter map. → Y p Proof: Let f : (X, H) → (Y, G) be an LM-G-filter map. Let B ∈ L be such that G (B) = 1. Then G(B)  p which implies H(f (B)) = 1. → p Y Conversely, let f : (X, H) → (Y, G ) be an LM-G-filter map. Let B ∈ L be such that p ← G(B)  p. Then G (B) = 1 and therefore H(f (B)) = 1. Hence the proof. → p Let (X, G) and (Y, H) be LM-G-filter spaces. Then it is easy to prove that f : (X, G ) → p → p (Y, H ) is an LM-G-filter map if f : (X, G) → (Y, H) is an LM-G-filter map. Thus () is a functor from LM-G to SLM -G. Moreover, it is easy to verify the following theorem. Theorem 5.6: () : LM-G → SLM -G is the right adjoint functor of i : SLM -G → LM-G. p p Theorem 5.7: SLM -G is a bicoreflective full subcategory of LM-G for all p ∈ pr(M). Proof: Since for a LM-G-filter space (X, G), id : (X, G ) → (X, G) is the SLM -G coreflection X p and it is bijective, SLM -G is a bicoreflective full subcategory of LM-G. Remark 5.8: The name strong p-level LM-G-filter space is justified since for an LM-G-filter p p space (X, G), id : (X, G ) → (X, G) is the SLM -G coreflection and (X, G ) is an LM-G-filter X p space which is stronger than (X, G) and takes only values 1 or p for any 1 = A ∈ L . As every right adjoint functor preserves limits we have the following corollaries. p p Corollary 5.9: Let (X, G) be LM-G-filter space and Y ⊆ X. Then G | = (G| ) . Y Y Corollary 5.10: Let {(X , G )} be a family of LM-G-filter spaces and X = X . Then j j j∈J j j∈J ( G ) = G . j j j∈J j∈J p p Corollary 5.11: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J 240 M. JOSE AND S. C. MATHEW It is easy to prove the following theorems. p p Theorem 5.12: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J Theorem 5.13: Let (X, G) be an LM-G-filter space and (Y, G/f ) be the LM quotient G-filter → X Y → p p → space of (X, G) with respect to the surjective mapping f : L → L . Then (G/f ) = G /f . Theorem 5.14: Let {(X , G )} be a family of LM-G-filter spaces, different X s be disjoint. Then j j j∈J p p ( G ) ≤ G . j j j∈J j∈J Remark 5.15: The above inequality in Theorem 5.14 cannot be replaced by equality. Let L = [1, 2], M = [0, 1]. For each j ∈ (0.5, 1], let X ={j} and G : L → M be defined by j j X X G (1 ) = 1and G (A ) = j for all other A ∈ L .Let X = X = (0.5, 1]. For A ∈ L such j X j j j j j j∈J 0.5 0.5 that A = 1 , G (A) = 1. But ( G ) (A) = 0.5. X j j∈J j∈J Since the objects in SLM -G are structured sets, SLM -G is a construct and it has certain p p categorical properties as proved in the following theorem. Theorem 5.16: SLM -G satisfies the following properties: (i) Existence of initial structures. (ii) Fibre-smallness. Proof: (i) Existence of initial structures:Let X be a nonempty set, {(X , G )} a family of strong j j j∈J p-level LM-G-filter spaces and {f : L → (X , G )} a family of maps. Then proceed- j j j j∈J ing as in Theorem 4.16, it is easy to verify that G : L → M defined by G(A) = {G (P ); f (P ) = B } is an strong p-level LM-G-filter on X such j j j j i B ≤A i∈I j∈J i∈I i that the source {f : (X, G) → (X , G )} is initial. j j j j∈J (ii) Fibre-smallness: From Corollary 5.11 and Theorem 5.12 it is clear that for any nonempty set X, the set of all strong p-level LM-G-filter spaces with underlying set X is a complete lattice with greatest element G defined by G(A) = 1 for all A ∈ L . Therefore SLM -G is fibre small. Remark 5.17: SLM -G is not topological. Let X ={x}, L = M = [0, 1]. Then L ={α ; α ∈ p X [0, 1]}. G and G defined by 1 2 1if α ∈ [0.8, 1], 1if α ∈ [0.7, 1], X X G (α ) = G (α ) = 1 X 2 X 0.3 if α ∈ [0, 0.8) 0.3 if α ∈ [0, 0.7) X X are strong 0.3-level LM-G-filters on X. Hence terminal separator property doesn’t hold for SLM -G. Therefore SLM -G is not topological. p p FUZZY INFORMATION AND ENGINEERING 241 6. Conclusion The study has identified some subcategories of the category of LM-G-filter spaces and obtained their categorical connections with LM-G. The study has introduced weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces and studied certain categorical properties of these spaces. The study identified WLM -G, the category of weak r-level LM- G-filter spaces as an isomorphism-closed bireflective full subcategory of LM-G, the category of LM-G-filter spaces. It is also proved that SLM -G, the category of strong p-level LM-G-filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G. The study has also identified certain level decompositions of LM-G-filter spaces and derived certain properties of the associated L-pre G-filter spaces. Disclosure statement No potential conflict of interest was reported by the authors. Funding The first author wishes to thank (Council of Scientific and Industrial Research, India) CSIR, India, for giving financial assistance under the Senior Research Fellowship awarded by order No. 08/528(0004)/2019-EMR-1 dated 08/04/2021. Notes on contributors Merin Jose received her M.Sc Degree from Cochin University of Science & Technology (CUSAT). She is currently doing Ph.D at Department of Mathematics, St. Thomas College Palai. Her research interest include fuzzy topology, fuzzy filter and category theory. Email: merinjmary@gmail.com Sunil C. Mathew received his M.Sc Degree from C.M.S College Kottayam and Ph.D from Mahatma Gandhi University, Kottayam. He held the position of Associate Professor at Department of Mathe- matics, St. Thomas College Palai. Currently, he is the Principal of Deva Matha College Kuravilangad. His research interest include fuzzy topology and graph theory. Email: sunilcmathew@gmail.com References [1] Lowen R. Convergence in fuzzy topological spaces. General Topolog Appl. 1979;10(2):147–160. [2] Burton MH, Muraleetharan M, Gutiérrez García J. Generalized filters 2. Fuzzy Sets Syst. 1999;106(3):393–400. [3] Höhle U, Šostak AP. Axiomatic foundations of fixed-basis fuzzy topology. In: Höhle U, Rodabaugh SE, editors. Mathematics of fuzzy sets: logic, topology and measure theory. Boston/Dordrecht/London: Kluwer; 1999. p. 123–272. [4] Kim YC, Ko JM. Images and preimages of L-filterbases. Fuzzy Sets Syst. 2006;157(14):1913–1927. [5] Jäger G. A note on stratified LM-filters. Iran J Fuzzy Syst. 2013;10:135–142. [6] Abbas SE, Aygün H, Cetkin V. On stratified L-filter structure. Int J Pure Appl Math. 2012;78(8):1221–1239. [7] Jose M, Mathew SC. Generalization of LM-Filters: sum, subspace, product, quotient and stratifi- cation, Communicated. [8] Jose M, Mathew SC. Catalyzed LM-G-filter spaces. J Intel Fuzzy Syst. 2022;1–11. [9] Jose M, Mathew SC. On the categorical connections of L-G-filter spaces and a Galois correspon- dence with L-fuzzy pre-proximity spaces. New Math Nat Comput. 2022. 242 M. JOSE AND S. C. MATHEW [10] Adàmek J, Herrlich H, Strecker GE. Abstract and concrete categories. New York: John Wiley and Sons; 1990. [11] Preuss G. Theory of topological structures, an approach to categorical topology. Dordrechet: Reidel; 1988. [12] Liu YM, Luo MK. Fuzzy topology. London: World Scientific Co.; 1997. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

On the Categories of Weak and Strong LM-G-Filter Spaces

Fuzzy Information and Engineering , Volume 14 (2): 15 – Apr 3, 2022

On the Categories of Weak and Strong LM-G-Filter Spaces

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In this paper, the authors introduce the notion of weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces and discuss certain properties of these spaces. The study identifies -G, the category of weak r-level LM-G-filter spaces as an isomorphism-closed bireflective full subcategory of LM-G, the category of LM-G-filter spaces. It is also proved that -G, the category of strong p-level LM-G-filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G. Moreover,...
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FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 228–242 https://doi.org/10.1080/16168658.2022.2057065 a b Merin Jose and Sunil C. Mathew a b Department of Mathematics, St. Thomas College Palai, Kottayam, Kerala, India; Department of Mathematics, Deva Matha College Kuravilangad, Kottayam, Kerala, India ABSTRACT ARTICLE HISTORY Received 2 June 2021 In this paper, the authors introduce the notion of weak r-level LM-G- Accepted 20 March 2022 filter spaces and strong p-level LM-G-filter spaces and discuss certain properties of these spaces. The study identifies WLM -G, the category KEYWORDS of weak r-level LM-G-filter spaces as an isomorphism-closed bireflec- Weak r-level LM-G-filter tive full subcategory of LM-G, the category of LM-G-filter spaces. It is spaces; strong p-level also proved that SLM -G, the category of strong p-level LM-G-filter p LM-G-filter spaces; initial spaces is an isomorphism-closed bicoreflective full subcategory of structures; fibre smallness; reflective subcategory; LM-G. Moreover, level decompositions of LM-G-filter spaces are stud- coreflective subcategory ied and some properties of the associated L-pre G-filter spaces are obtained. 2020 MATHEMATICS SUBJECT CLASSIFICATIONS 54A20; 18A40; 06D22 1. Introduction The study of filter and topology is highly interrelated. Because of the great importance of filter theory in topology, the concept of filters has been extended to different types of fuzzy filters. In 1977, Lowen [1] introduced the concept of filters in I and called them prefilters. Later in 1999, Burton et al. [2] introduced the concept of generalised filters as a map from 2 to I. Subsequently, Höhle and Šostak [3] developed the notion of L-filters and stratified L-filters on a complete quasi-monoidal lattice. In 2006, Kim et al. [4] introduced the notion of L-filter based on a strictly two-sided, commutative quantale lattice L and identified two types of images and preimages of L-filter bases. Later, in 2013 Jäger [5] developed the the- ory of stratified LM-filters which generalises the theory of stratified L-filters by introducing stratification mapping, where L and M are frames. In [6], Abbas et al. investigated strati- fied L-filter structure where L is a strictly two-sided, commutative quantale lattice. In [7], the authors disproved certain known theorems on L-filters in [6] and the rectification of the same led to the introduction of LM-G-filter spaces where those results hold good. More- over, an LM-filter is constructed from a given LM-G-filter and a categorical relationship is established between them in [7]. Recent studies on LM-G-filter spaces can be found in [8,9]. In this paper, we introduce the concept of weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces. It is proved that WLM -G, the category of weak r-level LM-G- filter spaces is an isomorphism-closed bireflective full subcategory of LM-G, the category CONTACT Sunil C. Mathew sunilcmathew@gmail.com © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 229 of LM-G-filter spaces. It is also proved that SLM -G, the category of strong p-level LM-G- filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G.Wehavealso constructed L-pre G-filter spaces by level decompositions of LM-G-filter spaces and studied their properties. 2. Preliminaries Throughout this paper, X stands for a non empty set, L and M stand for completely distribu- tive 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 the various notions of category theory, the readers can refer to [10,11]. Definition 2.1 ([12]): 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 ([12]): 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 [12] 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. Theorem 2.3 ([12]): Let L be a lattice, an order reversing involution on L, then (1) co-pr(L) = pr(L) (2) pr(L) =co-pr(L) Definition 2.4 ([7]): An LM-G-filter on a set X is defined to be a mapping G : L → M satisfying, G1: G(1 ) = 1; G2: For every A, B ∈ L such that A ≤ B, G(A) ≤ G(B); G3: For every A, B ∈ L , G(A ∧ B) ≥ G(A) ∧ G(B). The pair (X, G) is called an LM-G-filter space. When M ={0, 1}, it is called an L-pre G-filter space and when L ={0, 1},itiscalledan M-fuzzifying G-filter space. If G and G are two LM-G-filters on X such that G (A) ≥ G (A) for all A ∈ L , then we say 1 2 2 1 (X, G ) is weaker than (X, G ) and (X, G ) is stronger than (X, G ). 1 2 2 1 Remark 2.5: In addition to the above axioms, if G4: G(0 ) = 0isalsosatisfied, then (X, G) becomes an LM-filter space. 230 M. JOSE AND S. C. MATHEW → X Y Definition 2.6 ([7]): Let (X, G ) and (Y, G ) be LM-G-filter spaces. A map f : L → L is 1 2 ← Y called an LM-G-filter map if G (f (B)) ≥ G (B) for all B ∈ L . 1 2 → X Y Definition 2.7 ([7]): Let (X, G ) and (Y, G ) be LM-G-filter spaces. A map f : L → L is 1 2 → X called an LM-G-filter preserving map if G (f (A)) ≥ G (A) for all A ∈ L . 2 1 L-pre G-filter maps and L-pre G-filter preserving maps in L-pre G-filter spaces are defined analogously. Notation 2.8 ([7]): Let LM-G denotes the category of LM-G-filter spaces where morphisms are LM-G-filter maps and LM-G(X) denotes the lattice of LM-G-filters on X. Definition 2.9 ([7]): Let (X, G) be an LM-G-filter space and Y ⊆ X. Then the LM-G-filter, G| X Y defined on Y by (G| )(B) = {G(A)|A ∈ L , A| = B} for all B ∈ L is called the subspace of Y Y (X, G). Notation 2.10: Throughout this paper,  stands for finite intersection. Definition 2.11 ([7]): Let {(X , G )} be a family of LM-G-filter spaces, X = X and j j j∈J j j∈J p : X → X be the projection map. Then the product of {(X , G )} is defined as j j j j j j∈J j∈J X ← X G(A) = {G (D); D ∈ L , p (D) = B } for all A ∈ L . The product space j i B ≤A i∈I j∈J j i∈I i is denoted by (X, G ). j∈J Definition 2.12 ([7]): Let (X, G) be an LM-G-filter space and f : X → Y be a surjective map- → → ← Y ping. Then the LM-G-filter, G/f defined on Y by G/f (B) = G(f (B)) for all B ∈ L is called quotient LM-G-filter of G with respect to f. Definition 2.13 ([7]): Let {(X , G )} be a family of LM-G-filter spaces, X s be pairwise j j j∈J disjoint and X = X . Then the LM-G-filter, G defined on X by G (A) = j j j j∈J j∈J j∈J G (A| ) for all A ∈ L is called sum LM-G-filter of {G } . j X j j∈J j∈J j Definition 2.14 ([11]): A category C is said to be topological if the following conditions are satisfied: (i) Existence of initial structures:Foranyset X,anyfamily ((X , ξ )) of C-objects indexed i i i∈I by a class I and any family (f : X → X ) of maps indexed by I there exists a unique i i i∈I C-structure ξ on X which is initial with respect to (f : X → X ) in the sense that for i i i∈I any C-object (Y, η),amap g : (Y, η) → (X, ξ) is a C-morphism iff for every i ∈ I the composite map f og : (Y, η) → (X , ξ ) is a C-morphism. i i i (ii) Fibre-smallness: For any set X,the class {ξ |(X, ξ) is a C-object} of all C-structures with underlying set X, C-fibre of X,isaset. (iii) Terminal separator property: For any set X with cardinality atmost one, there exists exactly one C-structure on X. FUZZY INFORMATION AND ENGINEERING 231 3. Level Decompositions of LM-G-Filter Spaces In this section, we study level decompositions of LM-G-filter spaces with respect to ≥ relation and relation and study the properties of associated L-pre G-filter spaces. It is easy to prove the following theorem which associates a family of L-pre G-filter spaces to a given LM-G-filter space. Theorem 3.1: Let (X, G) be an LM-G-filter space. Then for each α ∈ M, G ={A ∈ L ; G(A) ≥ (α) α} is an L-pre G-filter on X. Remark 3.2: It is also easy to verify that if (X, G) is an LM-filter space, then G is an L-pre (α) filter for all α> 0. Remark 3.3: The L-pre G-filter G is called α-level decomposition of the LM-G-filter G with (α) respect to ≥ relation. Let L-PG denotes {G ; α ∈ M} (i.e. the set of all associated L-pre G- (G) (α) filters of an LM-G-filter space (X, G) with respect to ≥ relation) and L-PF denotes {F ; α ∈ (F) (α) M} (i.e. the set of all associated L-pre filters of an LM-filter space (X, F) with respect to ≥ relation). Theorem 3.4: Let G be an LM-G-filter on X. Then L-PG is a complete lattice. (G) Proof: Let {G ; α ∈ M ⊆ M} be an arbitrary collection in L-PG . Then it is easy to (α) 1 (G) observe that G = G .Also L = G ≥ G for every α ∈ M. Therefore (α) ( α) (0) (α) α∈M α∈M L-PG is a complete lattice. (G) Remark 3.5: Proceeding in the same way, it is easy to prove that given an LM-filter space (X, F), L-PF is a complete meet semilattice. (G) The following theorem gives an expression for join of an arbitrary collection of L-pre G-filters in L-PG . (G) Theorem 3.6: Let (X, G) be an LM-filter space and {G ; α ∈ M ⊆ M} be an arbitrary col- (α) 1 lection of L-pre G-filters in L-PG . Then G = G where K ={k ∈ M; G ⊇ (G) (α) ( k) (k) α∈M k∈K G ∀ α ∈ M }. (α) 1 Proof: Since L-PG is a complete lattice and G ⊇ G for every k ∈ K, (G) (k) (α) α∈M G = G .Itisclear that G ⊆ G . The reverse inequality is (α) (k) ( k) (k) α∈M k∈K k∈K k∈K obvious since, for every A ∈ L , A ∈ G ⇒ G(A) ≥ k for every k ∈ K. Therefore A ∈ (k) k∈K G . Hence G = G . Therefore G = G . ( k) ( k) (k) (α) ( k) k∈K α∈M k∈K k∈K k∈K The following theorem relates an L-pre G-filter, G in L-PG with the collection of all (α) (G) G where β ∈ β (α). (β) Theorem 3.7: Let (X, G) be an LM-G-filter space. Then (i) G ⊆ G whenever β  α. (α) (β) (ii) G = G where β ∈co-pr(M). (α) (β) βα 232 M. JOSE AND S. C. MATHEW Proof: Proof of (i) is trivial. Let A ∈ G (β) βα ⇒ G(A) ≥ β ∀β  α ⇒ G(A) ≥ β = α βα ⇒ A ∈ G (α) Therefore, G = G where β ∈ co-pr(M). (α) (β) βα Given a descending family of L-pre G-filters on X, we can construct an LM-G-filter on X as follows: Theorem 3.8: Let {G ; α ∈co-pr (M)} be a family of L-pre G-filters on X such that G ⊇ (α) (α ) G whenever α  α . Then G : L → M defined by G(A) = {α ∈ co-pr(M); A ∈ G } is an (α ) 1 2 (α) LM-G-filter on X. Moreover if G = G , then G = G . (α) (β) (α) (α) βα Proof: Since 1 ∈ G for all α, G(1 ) = 1. Let A, B ∈ L such that A ≤ B. Therefore, if X (α) X A ∈ G for some α, then B ∈ G . Hence G(A) ≤ G(B). Now let A, B ∈ L such that (α) (α) G(A) = p and G(B) = q.Let N ={α ∈ co-pr(M); A ∈ G } and N ={α ∈ co-pr(M); B ∈ 1 (α) 2 G }. Then p ∧ q = [ α] ∧ [ β] = (α ∧ β) = l where each (α) α∈N β∈N α∈N ,β∈N lα∧β 1 2 1 2 l  α ∧ β for some α ∈ N and β ∈ N . For any l  α ∧ β, l  α and l  β. Therefore A, B ∈ 1 2 G . Hence A ∧ B ∈ G for all l. Therefore G(A ∧ B) ≥ G(A) ∧ G(B). Hence G is an LM-G-filter (l) (l) on X. Let G = G . It is clear that G ⊆ G . The reverse inequality is obtained as (α) (β) (α) (α) βα A ∈ G ⇒ G(A) ≥ α = β where β ∈ co-pr(M) (α) βα ⇒ β  α ≤ G(A) = {k ∈ co-pr(M); A ∈ G }∀β  α (k) ⇒∃k ∈ co-pr(M); A ∈ G such that k ≥ β ⇒ G ⊆ G (β) (k) ⇒ A ∈ G ∀β  α (β) ⇒ A ∈ G = G (β) (α) βα Corollary 3.9: Let G and H be two LM-G-filters on X. Then G = H if and only if G = H for (α) (α) all α ∈ co-pr(M). Remark 3.10: Theorems 3.7 and 3.8 are valid in the case of LM-filters and L-prefilters also. Given an LM-G-filter space (X, G), we can associate a family of L-pre G-filters with respect to as follows: FUZZY INFORMATION AND ENGINEERING 233 Figure 1. The diamond type lattice. (α) X Theorem 3.11: Let (X, G) be an LM-G-filter space. Then for each α ∈ M, G ={A ∈ L ; α G(A)} is an L-pre G-filter on X. (α) (α) X Proof: G(1 ) = 1and α  1. Therefore 1 ∈ G for each α ∈ M.Let A ∈ G and B ∈ L be X X (α) such that A ≤ B. Therefore G(A) ≤ G(B) and α  G(A). This implies α  G(B). Hence B ∈ G . (α) Let A, B ∈ G . This implies α  G(A) and α  G(B). This implies there exists S ⊆ M such that S ≤ G(A) but for all s ∈ S , s  α andthereexists S ⊆ M such that S ≤ G(B) but 1 1 1 1 2 2 for all s ∈ S , s  α. Therefore, S = S S with S ≤ G(A) G(B) ≤ G(A ∧ B), but for 2 2 2 1 2 (α) (α) all s ∈ S, s  α. Therefore A ∧ B ∈ G . Hence G is an L-pre G-filter on X. (α) X Remark 3.12: Let (X, G) be an LM-filter space. Then for each α ∈ M, G ={A ∈ L ; α G(A)} need not be an L-prefilter on X. Let L = [0, 1], M be the product lattice [0, 1] , X ={x, y} and G be an LM-filter on X. Then (α) (α) for α = (0, 0.2) ∈ M, (0, 0.2)  0 = G(0 ) which implies 0 ∈ G . Hence G is not a pre X X L-filter on X. (α) Remark 3.13: The L-pre G-filter G is called α-level decomposition of the LM-G-filter G (G) (α) with respect to relation. Let L-PG denotes {G ; α ∈ M} (i.e. the set of all associated L-pre G-filters of an LM-G-filter space (X, G) with respect to relation). (G) Remark 3.14: Given an LM-G-filter space (X, G), L-PG is not a lattice. Let X ={x} and X X L = M be the lattice shown in Figure 1. Then L ={0 , α , β , γ ,1 }. G : L → M defined X X X X X (0) (α) by G(α ) = α for all α ∈ L is an LM-G-filter on X. Then G ={α , β , γ ,1 }, G = X X X X X (β) (γ ) (1) (α) (β) {β , γ ,1 }, G ={α γ ,1 }, G ={1 }, G ={1 }. Therefore G ∧ G ={γ ,1 } X X X X X X X X X X (α) (G) whichcannotbeexpressedas G for any α ∈ M. Hence L-PG is not a lattice. (α) (G) The following theorem relates a L-pre G-filter, G in L-PG with the collection of all (β) G where β ∈ pr(L) such that β ≺ α. Theorem 3.15: Let (X, G) be an LM-G-filter space. Then, (α) (β) (i) G ⊆ G whenever β ≺ α. (α) (β) (ii) G = G where β ∈ pr(L). β≺α 234 M. JOSE AND S. C. MATHEW Proof: Proof of (i) is trivial. (α) (β) (β) It is clear that G ⊆ G .Let A ∈ G where β is prime. This implies β β≺α β≺α G(A) for all β ≺ α.If α G(A), then there exists p ∈ pr(L) such that α p G(A) which is (α) a contradiction. Therefore A ∈ G . Given a descending family of L-pre G-filters on X, we can construct an LM-G-filter on X as follows: (α) (α ) (α ) 1 2 Theorem 3.16: Let {G ; α ∈ pr(M)} be a family of L-pre G-filters on X such that G ⊇ G X (α) whenever α ≺ α . Then G : L → M defined by G(A) = {α ∈ pr(M); A ∈ G } is an LM-G- 1 2 (α) (β) (α) (α) filter on X. Moreover if G = G , then G = G . β≺α (α) X Proof: Since 1 ∈ G for all α ∈ pr(M), G(1 ) = 1. Let A, B ∈ L such that A ≤ B.So B ∈ X X (α) (α) G for some α implies A ∈ G . Hence G(A) = (α) α ≤ (α) α = G(B). Now let A∈G B∈G X (α) A, B ∈ L such that G(A) = p and G(B) = q.Let N ={α ∈ pr(M); A ∈ G } and N ={α ∈ 1 2 (α) pr(M); B ∈ G }. Then G(A) ∧ G(B) = p ∧ q = [ (α ) α ] ∧ [ (β ) β ] = (β ) (α ∧ β ) = i j (α ) i j i j j A∈G i B∈G A∈G ,B∈G (l) l.Since α ∧ β ≤ α , β ,and l α ∧ β , l canbesuchthat A, B ∈ G or i j i j i j l α ∧β ,l∈pr(M) i j (l) (l) (l) (l) (l) A ∈ G , B ∈ G or A ∈ G , B ∈ G or A, B ∈ G . This implies G(A) ∧ G(B) = l ≤ l α ∧β i j p. Therefore G(A ∧ B) ≥ G(A) ∧ G(B). Hence G is an LM-G-filter on X. (p) A∧B∈G (α) (β) (α) Let G = G .Let A ∈ G . This implies β≺α (α) (β) (β) (q) α  G(A).Since G = G , A ∈ G for all β ≺ α. Therefore A ∈ G implies α  q β≺α (α) and G(A) = q. This implies α  G(A). Therefore A ∈ G . The reverse inequality is α q obtained as, (α) Let A ∈ G . This implies α  G(A) = (α ) α . This implies α  α for every α such i i i A∈G (α ) (α ) (α ) i i i that A ∈ G . Therefore, whenever A ∈ G then α  α .Thusif α α then A ∈ G .This i i (α ) (α) shows that A ∈ G = G . α ≺α (α) (α) Corollary 3.17: Let G and H be two LM-G-filters on X. Then G = H if and only if G = H for all α ∈ pr(M). [β] X Theorem 3.18: Let (X, G) be an LM-G-filter space. Then for each β ∈ M, G ={A ∈ L ; β G (A)} is an L-pre G-filter on X where G (A) = (G(A)) and is the order reversing involution in M. Proof: For a, b ∈ M,itiseasytoprove that a  b if and only if a b . This implies a [β] X  X  (β) b if and only if a  b . Therefore G ={A ∈ L ; β  G (A)}={A ∈ L ; β  G(A)}= G which is an L-pre G-filter on X. Theorem 3.19: Let (X, G) be an LM-G-filter space. Then; [α] [β] (i) G ⊆ G whenever α  β. [α] [β] (ii) G = G where β ∈ co-pr(M). βα Proof: Proof follows from Theorems 2.3 and 3.15.  FUZZY INFORMATION AND ENGINEERING 235 It is also easy to observe that, [α] [α ] Theorem 3.20: Let {G ; α ∈ co-pr(M)} be a family of L-pre G-filters on X such that G ⊇ [α ] X  [β] G whenever α  α . Then G : L → M defined by G (A) = {β ∈ copr(M); A ∈ G } is an 1 2 [α] [β] [α] [α] LM-G-filter on X. Moreover if G = G , then G = G . βα 4. Weak r-level LM-G-Filter Spaces In this section we introduce the concept of weak r-level LM-G-filter spaces and iden- tify WLM -G(the category of weak r-level LM-G-filter spaces) as an isomorphism-closed bireflective full subcategory of LM-G. Definition 4.1: An LM-G-filter space (X, G) which takes only the values 0 or r, where r ∈ M for any 1 = A ∈ L is called weak r-level LM-G-filter space. Let WLM -G denotes the category of weak r-level LM-G-filter spaces where objects are weak r-level LM-G-filter spaces and morphisms are LM-G-filter maps. Theorem 4.2: WLM -G is an isomorphism-closed full subcategory of LM-G for each r ∈ M. Proof: Let (X, G ) be a weak r-level LM-G-filter space, (Y, G ) be an LM-G-filter space and 1 2 → → Y f : (X, G ) → (Y, G ) be an isomorphism. Since f is an isomorphism, for any B ∈ L such 1 2 ← ← ← that B = 1 , f (B) = 1 .If G (f (B)) = 0, then it is clear that G (B) = 0. If G (f (B)) = r, Y X 1 2 1 then G (B) = r since f is an isomorphism. Therefore (Y, G ) is a weak r-level LM-G-filter 2 2 space. We give left adjoint of the inclusion functor i : WLM -G → LM-G and show that WLM -G r r is a bireflective full subcategory of LM-G. r X Lemma 4.3: Let (X, G) be an LM-G-filter space. Then G : L → M defined by 1 if A = 1 , ⎨ X G (A) = rifG(A) ≥ r and A = 1 , 0 otherwise is the largest weak r-level LM-G-filter smaller than G. Remark 4.4: If G is a weak r-level LM-G-filter, then G = G . Lemma 4.5: Let (X, G) be an LM-G-filter space and (Y, H) be a weak r-level LM-G-filter space. → → r Then f : (X, G) → (Y, H) is an LM-G-filter map if and only if f : (X, G ) → (Y, H) is an LM-G- filter map. → Y Proof: Let f : (X, G) → (Y, H) be an LM-G-filter map. Let B ∈ L be such that H(B) = r. ← r ← Then G(f (B)) ≥ H(B) = r which implies G (f (B)) ≥ r = H(B). → r Y Conversely, let f : (X, G ) → (Y, H) be an LM-G-filter map. Let B ∈ L be such that r ← ← H(B) = r. Then G (f (B)) ≥ r which implies G(f (B)) ≥ r. Hence the proof.  236 M. JOSE AND S. C. MATHEW → r Let (X, G) and (Y, H) be LM-G-filter spaces. Then it is easy to prove that f : (X, G ) → r → r (Y, H ) is an LM-G-filter map if f : (X, G) → (Y, H) is an LM-G-filter map. Thus () is a functor from LM-G to WLM -G. Moreover it is easy to verify the following theorem. Theorem 4.6: () : LM-G → WLM -G is the left adjoint functor of i : WLM -G → LM-G. r r Theorem 4.7: WLM -G is a bireflective full subcategory of LM-G for all r ∈ M. Proof: Since for a LM-G-filter space (X, G), id : (X, G) → (X, G ) is the WLM -G reflection X r and it is bijective, WLM -G is a bireflective full subcategory of LM-G. Remark 4.8: The name weak r-level LM-G-filter space is justified since for an LM-G-filter r r space (X, G), id : (X, G) → (X, G ) is the WLM -G reflection and (X, G ) is an LM-G-filter X r space which is weaker than (X, G) and takes only values 0 or r for any 1 = A ∈ L . As every left adjoint functor preserves colimits, we have the following corollaries: r r Corollary 4.9: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J Corollary 4.10: Let {(X , G )} be a family of LM-G-filter spaces, different X s be disjoint. Then j j j∈J r r ( G ) = G . j j j∈J j∈J Corollary 4.11: Let (X, G) be an LM-G-filter space and (Y, G/f ) be the LM-quotient G-filter → X Y → r r → space of (X, G) with respect to the surjective mapping f : L → L . Then (G/f ) = G /f . r r Theorem 4.12: Let (X, G) be LM-G-filter space and Y ⊆ X. Then G | = (G| ) . Y Y r r Y r X Proof: It is clear that G | ≤ (G| ) .For B ∈ L ,let (G| ) (B) = r.Let I ={A ∈ L ; A | = Y Y Y i i Y B}. Then G(A ) ≥ r.Since A | = B for all i ∈ I, ( A )| = B.Since G( A ) ≥ i i Y i Y i i∈I i∈I i∈I r r G(A ), G ( A ) = r. Hence G | (B) = r. i i Y i∈I i∈I It is easy to prove the following theorems. r r Theorem 4.13: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J Theorem 4.14: Let {(X , G )} be a family of LM-G-filter spaces and X = X . Then j j j∈J j j∈J r r G ≤ ( G ) . j j j∈J j∈J Remark 4.15: The above inequality in Theorem 4.14 cannot be replaced by equality. For example, let X ={x , x }, Y ={y , y } and L be the diamond lattice shown in Figure 2 and 1 2 1 2 M be the lattice shown in Figure 1. X × Y ={(x , y ), (x , y ), (x , y ), (x , y )}. 1 1 1 2 2 1 2 2 FUZZY INFORMATION AND ENGINEERING 237 Figure 2. The lattice M . Let the LM-G-filter on X, G be defined by A A A A A A A A A A A A A A A A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x 0000 αααα β β β β 1111 x 0 αβ 10 αβ 10 αβ 10 αβ 1 G (A ) 0000 αααα β β β β γ γ γ 1 1 i Let the LM-G-filter on Y, G be defined by B B B B B B B B B B B B B B B B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 y 0000 αααα β β β β 1111 y 0 αβ 10 αβ 10 αβ 10 αβ 1 G (B ) 0000 ββββ α α α α γ γ γ 1 2 i → → X×Y X Y Let p and p be the projection maps from L to L and L respectively. Since 1 2 G (0 ) = γ , ( G ) (0 ) = γ . But G (0 ) = 0. j X j X j X j∈{1,2} j∈{1,2} j∈{1,2} Since the objects in WLM -G are structured sets, WLM -G is a construct and it has certain r r categorical properties as proved in the following theorem. Theorem 4.16: WLM -G satisfies the following properties: (i) Existence of initial structures (ii) Fibre-smallness. Proof: (i) Existence of initial structures :Let X be a nonempty set, {(X , G )} a family of j j j∈J weak r-level LM-G-filter spaces and {f : X → (X , G )} a family of maps. Then it is easy j j j j∈J to verify that G : L → M defined by G(A) = {G (P ); f (P ) = B } j j j j i B ≤A i∈I j∈J i∈I i is a weak r-level LM-G-filter on X such that each f : (X, G) → (X , G ) is an LM-G-filter j j j map. Let (Y, H) be a weak r-level LM-G-filter space. If g : (Y, H) → (X, G) is an LM-G- filter map, then it is obvious that the composite map (f og) : (Y, H) → (X , G ) is an j j j LM-G-filter map for each j ∈ J.If (f og) : (Y, H) → (X , G ) are LM-G-filter maps for each j j j 238 M. JOSE AND S. C. MATHEW j ∈ J then for each A ∈ L , G(A) = {G (P ); f (P ) = B } j j j j i B ≤A i∈I j∈J i∈I i ← ← ← ← ≤ {H((f og) (P )); g (f (P )) = g (B )} j j j j i B ≤A i∈I j∈J i∈I i ≤ H(g (A)) Therefore g : (Y, H) → (X, G) is an LM-G-filter map. (ii) Fibre-smallness : From Corollary 4.9 and Theorem 4.13, it is clear that for any nonempty set X,thesetofallweak r-level LM-G-filter spaces with underlying set X is a complete lattice with greatest element G defined by G(1 ) = 1and G(A) = r for all other A ∈ L . Therefore WLM -G is fibre small. Remark 4.17: WLM -G is not topological. Let X ={x}, L = M = [0, 1]. Then L ={α ; α ∈ r X [0, 1]}. G and G defined by 1 2 ⎧ ⎧ 1if α = 1 , 1if α = 1 , ⎨ X X ⎨ X X G (α ) = 0.4 if α ∈ [0.8, 1), G (α ) = 0.4 if α ∈ [0.7, 1), 1 X X 2 X X ⎩ ⎩ 0if α ∈ [0, 0.8) 0if α ∈ [0, 0.7) X X are weak 0.4-level LM-G-filters on X. Hence terminal separator property doesn’t hold for WLM -G. Therefore WLM -G is not topological. r r 5. Strong p-level LM-G-Filter Spaces In this section, we introduce the concept of strong p-level LM-G-filter spaces and iden- tify SLM -G (the category of strong p-level LM-G-filter spaces) as an isomorphism-closed bicoreflective full subcategory of LM-G. Definition 5.1: An LM-G-filter space (X, G) which takes only the values 1 or p, where p ∈ pr(M) for any 1 = A ∈ L is called strong p-level LM-G-filter space. Let SLM -G denotes the category of strong p-level LM-G-filter spaces where objects are strong p-level LM-G-filter spaces and morphisms are LM-G-filter maps. Theorem 5.2: SLM -G is an isomorphism-closed full subcategory of LM-G for each p ∈ pr(M). Proof: Let (X, G ) be an strong p-level LM-G-filter space, (Y, G ) be an LM-G-filter space 1 2 → → and f : (X, G ) → (Y, G ) be an isomorphism. Since f is an isomorphism, for any B ∈ 1 2 Y → ← ← ← L , G (B) = G (f (f (B))) ≥ G (f (B). Therefore, if G (f (B)) = 1, then G (B) = 1. If 2 2 1 1 2 ← ← G (f (B)) = p, then G (B) ≤ G (f (B)) = p. Therefore (Y, G ) is an strong p-level LM-G- 1 2 1 2 filter space. We give right adjoint of the inclusion functor i : SLM -G → LM-G and show that SLM -G p p is a bicoreflective full subcategory of LM-G. FUZZY INFORMATION AND ENGINEERING 239 p X Lemma 5.3: Let (X, G) be an LM-G-filter space. Then G : L → M be defined by pifG(A) ≤ p, G (A) = 1 if G(A)  p is the smallest strong p-level LM-G-filter larger than G. p X p Proof: Clearly G (1 ) = 1. Let A, B ∈ L such that A ≤ B.If G (A) = 1, then G(A)  p which p p p X implies G(B)  p and hence G (B) = 1. Therefore G (A) ≤ G (B).If A, B ∈ L such that p p G (A) = G (B) = 1, then G(A)  p and G(B)  p.Since p is prime, G(A) G(B)  p. There- fore G(A ∧ B)  p and hence G (A ∧ B) = 1. Remaining part of the proof is trivial. Remark 5.4: If G is a strong p-level LM-G-filter, then G = G . Lemma 5.5: Let (X, H) be a strong p-level LM-G-filter space and (Y, G) be an LM-G-filter space. → → p Then f : (X, H) → (Y, G) is an LM-G-filter map if and only if f : (X, H) → (Y, G ) is an LM- G-filter map. → Y p Proof: Let f : (X, H) → (Y, G) be an LM-G-filter map. Let B ∈ L be such that G (B) = 1. Then G(B)  p which implies H(f (B)) = 1. → p Y Conversely, let f : (X, H) → (Y, G ) be an LM-G-filter map. Let B ∈ L be such that p ← G(B)  p. Then G (B) = 1 and therefore H(f (B)) = 1. Hence the proof. → p Let (X, G) and (Y, H) be LM-G-filter spaces. Then it is easy to prove that f : (X, G ) → p → p (Y, H ) is an LM-G-filter map if f : (X, G) → (Y, H) is an LM-G-filter map. Thus () is a functor from LM-G to SLM -G. Moreover, it is easy to verify the following theorem. Theorem 5.6: () : LM-G → SLM -G is the right adjoint functor of i : SLM -G → LM-G. p p Theorem 5.7: SLM -G is a bicoreflective full subcategory of LM-G for all p ∈ pr(M). Proof: Since for a LM-G-filter space (X, G), id : (X, G ) → (X, G) is the SLM -G coreflection X p and it is bijective, SLM -G is a bicoreflective full subcategory of LM-G. Remark 5.8: The name strong p-level LM-G-filter space is justified since for an LM-G-filter p p space (X, G), id : (X, G ) → (X, G) is the SLM -G coreflection and (X, G ) is an LM-G-filter X p space which is stronger than (X, G) and takes only values 1 or p for any 1 = A ∈ L . As every right adjoint functor preserves limits we have the following corollaries. p p Corollary 5.9: Let (X, G) be LM-G-filter space and Y ⊆ X. Then G | = (G| ) . Y Y Corollary 5.10: Let {(X , G )} be a family of LM-G-filter spaces and X = X . Then j j j∈J j j∈J ( G ) = G . j j j∈J j∈J p p Corollary 5.11: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J 240 M. JOSE AND S. C. MATHEW It is easy to prove the following theorems. p p Theorem 5.12: Let {(X, G )} be a family of LM-G-filter spaces. Then ( G ) = G . j j∈J j j j∈J j∈J Theorem 5.13: Let (X, G) be an LM-G-filter space and (Y, G/f ) be the LM quotient G-filter → X Y → p p → space of (X, G) with respect to the surjective mapping f : L → L . Then (G/f ) = G /f . Theorem 5.14: Let {(X , G )} be a family of LM-G-filter spaces, different X s be disjoint. Then j j j∈J p p ( G ) ≤ G . j j j∈J j∈J Remark 5.15: The above inequality in Theorem 5.14 cannot be replaced by equality. Let L = [1, 2], M = [0, 1]. For each j ∈ (0.5, 1], let X ={j} and G : L → M be defined by j j X X G (1 ) = 1and G (A ) = j for all other A ∈ L .Let X = X = (0.5, 1]. For A ∈ L such j X j j j j j j∈J 0.5 0.5 that A = 1 , G (A) = 1. But ( G ) (A) = 0.5. X j j∈J j∈J Since the objects in SLM -G are structured sets, SLM -G is a construct and it has certain p p categorical properties as proved in the following theorem. Theorem 5.16: SLM -G satisfies the following properties: (i) Existence of initial structures. (ii) Fibre-smallness. Proof: (i) Existence of initial structures:Let X be a nonempty set, {(X , G )} a family of strong j j j∈J p-level LM-G-filter spaces and {f : L → (X , G )} a family of maps. Then proceed- j j j j∈J ing as in Theorem 4.16, it is easy to verify that G : L → M defined by G(A) = {G (P ); f (P ) = B } is an strong p-level LM-G-filter on X such j j j j i B ≤A i∈I j∈J i∈I i that the source {f : (X, G) → (X , G )} is initial. j j j j∈J (ii) Fibre-smallness: From Corollary 5.11 and Theorem 5.12 it is clear that for any nonempty set X, the set of all strong p-level LM-G-filter spaces with underlying set X is a complete lattice with greatest element G defined by G(A) = 1 for all A ∈ L . Therefore SLM -G is fibre small. Remark 5.17: SLM -G is not topological. Let X ={x}, L = M = [0, 1]. Then L ={α ; α ∈ p X [0, 1]}. G and G defined by 1 2 1if α ∈ [0.8, 1], 1if α ∈ [0.7, 1], X X G (α ) = G (α ) = 1 X 2 X 0.3 if α ∈ [0, 0.8) 0.3 if α ∈ [0, 0.7) X X are strong 0.3-level LM-G-filters on X. Hence terminal separator property doesn’t hold for SLM -G. Therefore SLM -G is not topological. p p FUZZY INFORMATION AND ENGINEERING 241 6. Conclusion The study has identified some subcategories of the category of LM-G-filter spaces and obtained their categorical connections with LM-G. The study has introduced weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces and studied certain categorical properties of these spaces. The study identified WLM -G, the category of weak r-level LM- G-filter spaces as an isomorphism-closed bireflective full subcategory of LM-G, the category of LM-G-filter spaces. It is also proved that SLM -G, the category of strong p-level LM-G-filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G. The study has also identified certain level decompositions of LM-G-filter spaces and derived certain properties of the associated L-pre G-filter spaces. Disclosure statement No potential conflict of interest was reported by the authors. Funding The first author wishes to thank (Council of Scientific and Industrial Research, India) CSIR, India, for giving financial assistance under the Senior Research Fellowship awarded by order No. 08/528(0004)/2019-EMR-1 dated 08/04/2021. Notes on contributors Merin Jose received her M.Sc Degree from Cochin University of Science & Technology (CUSAT). She is currently doing Ph.D at Department of Mathematics, St. Thomas College Palai. Her research interest include fuzzy topology, fuzzy filter and category theory. Email: merinjmary@gmail.com Sunil C. Mathew received his M.Sc Degree from C.M.S College Kottayam and Ph.D from Mahatma Gandhi University, Kottayam. He held the position of Associate Professor at Department of Mathe- matics, St. Thomas College Palai. Currently, he is the Principal of Deva Matha College Kuravilangad. His research interest include fuzzy topology and graph theory. Email: sunilcmathew@gmail.com References [1] Lowen R. Convergence in fuzzy topological spaces. General Topolog Appl. 1979;10(2):147–160. [2] Burton MH, Muraleetharan M, Gutiérrez García J. Generalized filters 2. Fuzzy Sets Syst. 1999;106(3):393–400. [3] Höhle U, Šostak AP. Axiomatic foundations of fixed-basis fuzzy topology. In: Höhle U, Rodabaugh SE, editors. Mathematics of fuzzy sets: logic, topology and measure theory. Boston/Dordrecht/London: Kluwer; 1999. p. 123–272. [4] Kim YC, Ko JM. Images and preimages of L-filterbases. Fuzzy Sets Syst. 2006;157(14):1913–1927. [5] Jäger G. A note on stratified LM-filters. Iran J Fuzzy Syst. 2013;10:135–142. [6] Abbas SE, Aygün H, Cetkin V. On stratified L-filter structure. Int J Pure Appl Math. 2012;78(8):1221–1239. [7] Jose M, Mathew SC. Generalization of LM-Filters: sum, subspace, product, quotient and stratifi- cation, Communicated. [8] Jose M, Mathew SC. Catalyzed LM-G-filter spaces. J Intel Fuzzy Syst. 2022;1–11. [9] Jose M, Mathew SC. On the categorical connections of L-G-filter spaces and a Galois correspon- dence with L-fuzzy pre-proximity spaces. New Math Nat Comput. 2022. 242 M. JOSE AND S. C. MATHEW [10] Adàmek J, Herrlich H, Strecker GE. Abstract and concrete categories. New York: John Wiley and Sons; 1990. [11] Preuss G. Theory of topological structures, an approach to categorical topology. Dordrechet: Reidel; 1988. [12] Liu YM, Luo MK. Fuzzy topology. London: World Scientific Co.; 1997.

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Apr 3, 2022

Keywords: Weak r -level LM -G-filter spaces; strong p -level LM -G-filter spaces; initial structures; fibre smallness; reflective subcategory; coreflective subcategory; 54A20; 18A40; 06D22

References