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

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, level decompositions of LM-G-filter spaces are studied and some properties of the associated L-pre G-filter spaces are obtained.


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 X and called them prefilters. Later in 1999, Burton et al. [2] introduced the concept of generalised filters as a map from 2 X 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 theory 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 stratified 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. Moreover, 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 r -G, the category of weak r-level LM-Gfilter spaces is an isomorphism-closed bireflective full subcategory of LM-G, the category CONTACT Sunil C. Mathew sunilcmathew@gmail.com of LM-G-filter spaces. It is also proved that SLM p -G, the category of strong p-level LM-Gfilter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G. We have also constructed L-pre G-filter spaces by level decompositions of LM-G-filter spaces and studied their properties.

Preliminaries
Throughout this paper, X stands for a non empty set, L and M stand for completely distributive lattices with an order reversing involution. An element α ∈ L is called prime if α < 1 and 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 2.2 ([12]):
Let L be a complete lattice. Define a relation in L as follows: 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.

Definition 2.4 ([7]):
An LM-G-filter on a set X is defined to be a mapping G : L X → M satisfying, 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}, it is called an M-fuzzifying G-filter space.
If G 1 and G 2 are two LM-G-filters on X such that G 2 (A) ≥ G 1 (A) for all A ∈ L X , then we say (X, G 1 ) is weaker than (X, G 2 ) and (X, G 2 ) is stronger than (X, G 1 ).

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 2.7 ([7]): Let
L-pre G-filter maps and L-pre G-filter preserving maps in L-pre G-filter spaces are defined analogously.
Notation 2.10: Throughout this paper, stands for finite intersection.

Definition 2.11 ([7]):
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 The product space is denoted by (X, j∈J G j ).

Definition 2.13 ([7]):
Let {(X j , G j )} j∈J be a family of LM-G-filter spaces, X j s be pairwise disjoint and X = j∈J X j . Then the LM-G-filter, j∈J G j defined on X by

Definition 2.14 ([11]):
A category C is said to be topological if the following conditions are satisfied: (i) Existence of initial structures: For any set X, any family ((X i , ξ i )) i∈I of C-objects indexed by a class I and any family (f i : X → X i ) i∈I of maps indexed by I there exists a unique C-structure ξ on X which is initial with respect to (f i : X → X i ) i∈I in the sense that for (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, is a set. (iii) Terminal separator property: For any set X with cardinality atmost one, there exists exactly one C-structure on X.

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.

Remark 3.2:
It is also easy to verify that if (X, G) is an LM-filter space, then G (α) is an L-pre filter for all α > 0.
the set of all associated L-pre filters of an LM-filter space (X, F) with respect to ≥ relation).

Remark 3.5:
Proceeding in the same way, it is easy to prove that given an LM-filter space (X, F), L-PF (G) is a complete meet semilattice.
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
The following theorem relates an L-pre G-filter, Theorem 3.7: Let (X, G) be an LM-G-filter space. Then Given a descending family of L-pre G-filters on X, we can construct an LM-G-filter on X as follows: The reverse inequality is obtained as Given an LM-G-filter space (X, G), we can associate a family of L-pre G-filters with respect to as follows: The following theorem relates a L-pre G-filter, G (α) in L-PG (G) with the collection of all G (β) where β ∈ pr(L) such that β ≺ α.
, then there exists p ∈ pr(L) such that α p G(A) which is a contradiction. Therefore A ∈ G (α) .
Given a descending family of L-pre G-filters on X, we can construct an LM-G-filter on X as follows: The reverse inequality is obtained as, Let A ∈ G (α) . This implies α G(A) = A ∈G (α i ) α i . This implies α α i for every α i such that A ∈ G (α i ) . Therefore, whenever A ∈ G (α i ) then α α i . Thus if α α i then A ∈ G (α i ) . This shows that A ∈ α i ≺α G (α i ) = G (α) .

Proof: Proof follows from Theorems 2.3 and 3.15.
It is also easy to observe that,

Weak r-level LM-G-Filter Spaces
In this section we introduce the concept of weak r-level LM-G-filter spaces and identify WLM r -G(the category of weak r-level LM-G-filter spaces) as an isomorphism-closed bireflective full subcategory of LM-G.

Theorem 4.2: WLM r -G is an isomorphism-closed full subcategory of LM-G for each r ∈ M.
Proof: We give left adjoint of the inclusion functor i : WLM r -G → LM-G and show that WLM r -G is a bireflective full subcategory of LM-G.    = H(B).
Conversely, let f → : (X, G r ) → (Y, H) be an LM-G-filter map. Let B ∈ L Y be such that H(B) = r. Then G r (f ← (B)) ≥ r which implies G(f ← (B)) ≥ r. Hence the proof.
Let (X, G) and (Y, H) be LM-G-filter spaces. Then it is easy to prove that f → : H) is an LM-G-filter map. Thus () r is a functor from LM-G to WLM r -G. Moreover it is easy to verify the following theorem. Proof: Since for a LM-G-filter space (X, G), id X : (X, G) → (X, G r ) is the WLM r -G reflection and it is bijective, WLM r -G is a bireflective full subcategory of LM-G. reflection and (X, G r ) is an LM-G-filter space which is weaker than (X, G) and takes only values 0 or r for any 1 X = A ∈ L X .

Remark 4.8: The name weak r-level LM-G-filter space is justified since for an LM-G-filter
As every left adjoint functor preserves colimits, we have the following corollaries: Corollary 4.9: Let {(X, G j )} j∈J be a family of LM-G-filter spaces. Then ( j∈J G j ) r = j∈J G j r .

Proof: It is clear that
It is easy to prove the following theorems.  Let the LM-G-filter on X, G 1 be defined by Let the LM-G-filter on Y, G 2 be defined by Let p 1 → and p 2 → be the projection maps from L X×Y to L X and L Y respectively. Since Since the objects in WLM r -G are structured sets, WLM r -G is a construct and it has certain categorical properties as proved in the following theorem.  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, the set of all weak r-level LM-G-filter spaces with underlying set X is a complete lattice with greatest element G defined by G(1 X ) = 1 and G(A) = r for all other A ∈ L X . Therefore WLM r -G is fibre small.
are weak 0.4-level LM-G-filters on X. Hence terminal separator property doesn't hold for WLM r -G. Therefore WLM r -G is not topological.

Strong p-level LM-G-Filter Spaces
In this section, we introduce the concept of strong p-level LM-G-filter spaces and identify SLM p -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 X = A ∈ L X is called strong p-level LM-G-filter space.
Let SLM p -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 p -G is an isomorphism-closed full subcategory of LM-G for each p ∈ pr(M).
Proof: Let (X, G 1 ) be an strong p-level LM-G-filter space, (Y, G 2 ) be an LM-G-filter space and f → : We give right adjoint of the inclusion functor i : SLM p -G → LM-G and show that SLM p -G is a bicoreflective full subcategory of LM-G. Lemma 5.3: Let (X, G) be an LM-G-filter space. Then G p : L X → M be defined by is the smallest strong p-level LM-G-filter larger than G.
It is easy to prove the following theorems.
Theorem 5.12: Let {(X, G j )} j∈J be a family of LM-G-filter spaces. Then ( j∈J G j ) p = j∈J G j p .
Theorem 5.13: Let (X, G) be an LM-G-filter space and (Y, G/f → ) be the LM quotient G-filter space of (X, G) with respect to the surjective mapping f → : L X → L Y . Then (G/f → ) p = G p /f → .
Theorem 5.14: Let {(X j , G j )} j∈J be a family of LM-G-filter spaces, different X j s be disjoint. Then ( j∈J G j ) p ≤ j∈J G j p .
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 = {j} and G j : L X j → M be defined by G j (1 X j ) = 1 and G j (A j ) = j for all other A j ∈ L X j . Let X = j∈J X j = (0. 5,1]. For A ∈ L X such that A = 1 X , j∈J G 0.5 j (A) = 1. But ( j∈J G j ) 0.5 (A) = 0.5.
Since the objects in SLM p -G are structured sets, SLM p -G is a construct and it has certain categorical properties as proved in the following theorem.

Theorem 5.16: SLM p -G satisfies the following properties:
(i) Existence of initial structures.

Proof:
(i) Existence of initial structures: Let X be a nonempty set, {(X j , G j )} j∈J a family of strong p-level LM-G-filter spaces and {f j : L X → (X j , G j )} j∈J a family of maps. Then proceeding as in Theorem 4.16, it is easy to verify that G : L X → M defined by G(A) = i∈I B i ≤A i∈I j∈J {G j (P j ); f j ← (P j ) = B i } is an strong p-level LM-G-filter on X such that the source {f j : (X, G) → (X j , G j )} j∈J is initial. (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 X . Therefore SLM p -G is fibre small. are strong 0.3-level LM-G-filters on X. Hence terminal separator property doesn't hold for SLM p -G. Therefore SLM p -G is not topological.