L-fuzzy filters on complete residuated lattices

This paper is toward the establishment of relationships between L-fuzzy filters, L-fuzzy topological spaces and L-fuzzy pre-proximity spaces in complete residuated lattices. We have demonstrated the existence of functors between the categories of L-fuzzy filter spaces, L-fuzzy topological spaces and L-fuzzy pre-proximity spaces.

1 Introduction Ward and Dilworth (1939) introduced the notion of complete residuated lattice as a primitive concept which is highly useful for structure of truth value in many valued logic.Bělohlávek (2002) proved that fuzzy relations with truth values in complete residuated lattice are capable of modeling intelligent systems with insufficient and incomplete information.Höhle and Šostak (1999) used different algebraic structures (cqm, quantales, M V -algebra) of truth value to introduce concepts of L-fuzzy topologies.Further, these algebraic structures provided several directions of study in mathematics as well as in logic and L-fuzzy topologies (cf., Fang 2010;Fang and Yue 2010;Koguep et al. 2008;Kubiak 1985;Kubiak and Šostak 1997;Chen and Zhang 2010;Ramadan et al. 2015;Ramadan and Kim 2018;Ramadan et al. 2022;Rodabaugh and Klement 2003;Šostak 1985, 1989;Tiwari et al. 2018;Yue 2007;Zhang 2007;Ramadan 1992;Liang and Shi 2014).
Many authors studied the relationship between fuzzy topologies and L-filters.In 1977, Lowen (1979) developed the idea of filters in I X where I = [0, 1] is the unit interval of real numbers, called prefilters to discuss convergence in fuzzy topological spaces.In 1999, Burton et al. (1999) introduced the concept of generalized filters as a mapping from 2 X to I .Subsequently, Höhle and Šostak (1999) developed the notion of L-filters and stratified L-filters on a complete quasi-monoidal lattice.Later, in Jäger (2013) developed the B Ahmed A. Ramadan ahmed.ramadan@science.bsu.edu.eg 1 Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt theory of stratified L M-filters which generalizes the theory of stratified L-filters by introducing stratification mapping, where L and M are frames (cf., Ko 2018;Koguep et al. 2008;Ramadan 1997;Liu et al. 2017;Tonga 2011).In Ramadan (2003), the authors introduced the concept of smooth ideal as a mapping from I X to I which is the dual of a smooth filter (Ramadan 1997).
In this paper, we identify L-fuzzy topologies and Lfuzzy pre-proximities induced by L-fuzzy (prime) filters and study categorical relations between L-fuzzy (prime) filter spaces, L-fuzzy topological spaces and L-fuzzy preproximity spaces.The study obtains functors from the categories of L-fuzzy (prime) filter spaces, L-fuzzy topological spaces and L-fuzzy pre-proximity spaces.

Preliminaries
Definition 1 (Bělohlávek 2002;Hájek 1998;Höhle and Šostak 1999;Rodabaugh and Klement 2003;Turunen 1999) A complete residuated lattice is a pair (L, ) which satisfies the following conditions: (C1) (L, ≤, ∨, ∧, ⊥, ) is a complete lattice with the greatest element and the least element ⊥; The binary relation induces another binary operation → on L which satisfies: In this paper, we always assume that L = (L, ≤, ) is a complete residuated lattice unless otherwise specified.
Lemma 1 Let L be a complete residuated lattice.For each x, y, z, x i , y i , w ∈ L, i ∈ Γ , we have the following properties: L is said to satisfy the double negation law if for any x ∈ L, (x → ⊥) → ⊥ = x.In the following, we use x * to denote x → ⊥.Furthermore, for any x, y ∈ L, we define x ⊕ y = (x * y * ) * .
Lemma 2 If L satisfies the double negation law, then it satisfies moreover: Definition 2 (Bělohlávek 2002;Rodabaugh and Klement 2003) Let X be a set.A mapping R An L-fuzzy relation on X is called an L-fuzzy pre-order if it is reflexive and transitive.
All algebraic operation on L can be extended pointwise to L X Goguen (1967).For f , g ∈ L X , we denote , otherwise.
Lemma 3 (Bělohlávek 2002;Fang 2010;Fang and Yue 2010) Let X be a nonempty set, define a binary mapping S : Then, for each f , g, f i , g i , h, l ∈ L X , i ∈ Γ , the following properties hold: Definition 3 (Adámek et al. 1990) A pair (C, U ) is said to be a concrete category if C is a category and U : C → Set is a faithful functor (or a forgetful functor).For each Cobject X , U (X ) is the underlying set of X .Thus, all objects in a concrete category can be taken as structured set.We write C for (C, U ), if the concrete functor is clear.Categories presented in this paper are concrete categories.A concrete functor between two concrete categories (C, U ) and G only changes the structures on the underlying sets.Hence, in order to define a concrete functor G : C → D, we only consider the following two requirements.First, we assign to Definition 4 (Höhle and Šostak 1999; Rodabaugh and Klement 2003) A mapping T : L X → L is called L-fuzzy topology on X if it satisfies the following conditions: The pair (X , T ) is called an L-fuzzy topological space.
An L-fuzzy topological space is called Definition 5 (Chen and Zhang 2010; Xiu and Li 2019) Let (X , T X ) and (Y , T Y ) be two L-fuzzy topological spaces and ϕ : X → Y be a mapping.Then, D T (ϕ) defined by is the degree to which the map ϕ is an L F-continuous map. If which is exactly the definition of L F-continuous map between L-fuzzy topological spaces.
The category of L-fuzzy topological spaces with L Fcontinuous mappings as morphisms is denoted by L-FTOP.Write AL-FTOP for the full subcategory of L-FTOP composed of objects of all Alexandrov L-fuzzy topological spaces.
Definition 6 (Ko 2018;Rodabaugh and Klement 2003) An L-fuzzy pre-filter on a set X is defined to be a mapping F : Definition 7 Let (X , F X ) and (Y , F Y ) be two L-fuzzy filter spaces and ϕ : X → Y be a mapping.Then, D F (ϕ) defined by is the degree to which the map ϕ is an L F-filter map. If which is exactly the definition of L F-filter map between L-fuzzy filter spaces.
Remark 2 In addition to the above axioms, if (LF4) F( X ) = , then (X , F) is called L-fuzzy prime filter space.
The category of L-fuzzy (prime) filter spaces with L Ffilter mappings as morphisms is denoted by LF(P-LF).Write A-LF (AP-LF) for the full subcategory of LF(P-LF) composed of objects of all Alexandrov L-fuzzy (prime) filter spaces.

The relationships between L-fuzzy (prime) filter spaces and topological spaces
From the following theorems, we obtain the L-fuzzy topological spaces induced by an L-fuzzy prime filter spaces Theorem 1 Let F be an L-fuzzy (prime) filter on X and L satisfies the double negation law.Define T (1) F : L X → L as follows: 123

Hence, T
(1) F is an L-fuzzy topology on X . (2) (3) For each family = T (1) Theorem 2 Let (X , F X ) and (Y , F Y ) be L-fuzzy (prime) filter spaces and L satisfies the double negation law.Let By Theorems 1 and 2, we obtain the following corollary: If we still write for the restriction of the functor Υ : P-LF → L-FTOP to the full subcategory AP-LF, then by Theorem 1, Υ : AP-LF → AL-FTOP forms a functor.

123
Theorem 3 Let F be an L-fuzzy (prime) filter on X .Define T (2) F : L X → L as follows: (T3) For each family { f i : i ∈ Γ }, we have

Hence, T (2)
F is an L-fuzzy topology on X .
(2) T (2) (3) For each family Theorem 4 Let (X , F X ) and (Y , F Y ) be L-fuzzy (prime) filter spaces and ϕ : X → Y be a mapping, then D From the above theorem, we deduce that if ϕ : By Theorems 3 and 4, we obtain the following corollary: Corollary 2 Ω : P-LF → L-FTOP is a functor defined by If we still write for the restriction of the functor Ω : P-LF → L-FTOP to the full subcategory AP-LF, then by Theorem 3, Ω : AP-LF → AL-FTOP forms a functor.

Theorem 5 Let F be an L-fuzzy prime filter on X . Define a mapping T
(3) Then, (1) (X , T (3)

The relationships between L-fuzzy pre-proximities and L-fuzzy filters
In this section, we introduce the relationship between Lfuzzy pre-proximity spaces and L-fuzzy filter spaces.
Definition 8 An L-fuzzy pre-proximity on X is a mapping Definition 9 Let (X , δ X ) and (Y , δ Y ) be two L-fuzzy preproximities and ϕ : X → Y be a mapping.Then, D δ (ϕ) defined by which is exactly the definition of L Fproximity map between L-fuzzy pre-proximities.
The category of L-fuzzy pre-proximity spaces with L Fproximity mappings as morphisms is denoted by L-PROX.Write AL-PROX for the full subcategory of L-PROX composed of objects of all Alexandrov L-fuzzy pre-proximity spaces.
In the sequel, we assume that L satisfies the double negation law.
Theorem 7 Let L be idempotent, δ be an L-fuzzy preproximity.Define a mapping F k δ : L X −→ L as follows: Now, let F(X ) be the family of all L-fuzzy prime filter and P(X ) be the family of all L-fuzzy pre-proximities on X .
Theorem 8 Let L be idempotent, H : P(X ) × F(X ) → F(X ) be a mapping defined as follows: Then, we have the following properties: (2) Let f ∈ L X such that f = ⊥ X , then Conversely, Theorem 9 Let F be an L-fuzzy prime filter on X such that F(g) ≤ g(x) for each x ∈ X and g ∈ L X .Define a mapping Then, δ F is an L-fuzzy pre-proximity on X .Moreover, if F is discrete (resp., Alexandrov ), then so is δ F .
Theorem 10 Let (X , F X ) and (Y , F Y ) be L-fuzzy filter spaces and ϕ : X → Y be a mapping.Then, D F (ϕ) ≤ D δ F (ϕ).
Proof For every f , g ∈ L Y , we have By Theorems 9 and 10, we obtain the following corollary: If we still write for the restriction of the functor Φ : P-LF → L-PROX to the full subcategory AP-LF, then by Theorem 9, Δ : AP-LF → AL-PROX forms a functor.
Let L-FRR be a category with object (X , R X ), where R X is a reflexive L-fuzzy relation with an order preserving map ϕ : Theorem 11 Let R X be a reflexive L-fuzzy relation.Define a mapping F x R : L X → L as follows: Then, (1) F x R is an Alexandrov L-fuzzy filter on X , 123 By Theorem 11, we obtain the following corollary: Corollary 5 Ψ : L-FRR → A-LF is a functor defined by Ψ (X , F x ) = (X , δ F x ), Ψ (ϕ) = ϕ.As an information system and an extension of Pawlak's rough set (Pawlak 1982(Pawlak , 1991)), we give the following example for L-fuzzy pre-proximities and L-fuzzy filters.
Example 1 (1) Define F 1 : Hence, F 1 is Alexandrov L-fuzzy filter on X .Since (2) Define F 2 : L X → L as F 2 ( f ) = f (x).Hence, F 1 is a discrete and Alexandrov L-fuzzy filter on X .By Theorem 9, we have .