EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 949-968 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the aspects of enriched lattice-valued topological groups and closure of lattice-valued subgroups T M G Ahsanullah1,∗, Fawzi Al-Thukair1 1 Department of Mathematics, King Saud University, Riyadh, Saudi Arabia. Dedicated to Professor John N. Mordeson on the occasion of his 87th birthday Abstract. Starting with L as an enriched cl-premonoid, in this paper, we explore some categor- ical connections between L-valued topological groups and Kent convergence groups, where it is shown that every L-valued topological group determines a well-known Kent convergence group, and conversely, every Kent convergence group induces an L-valued topological group. Considering an L-valued subgroup of a group, we show that the category of L-valued groups, L-GRP has ini- tial structure. Furthermore, we consider a category L-CLS of L-valued closure spaces, obtaining its relation with L-valued Moore closure, and provide examples in relation to L-valued subgroups that produce Moore collection. Here we look at a category of L-valued closure groups, L-CLGRP proving that it is a topological category. Finally, we obtain a relationship between L-GRP and L-TransTOLGRP, the category of L-transitive tolerance groups besides adding some properties of L-valued closures of L-valued subgroups on L-valued topological groups. 2020 Mathematics Subject Classifications: 03E72, 20N25, 18B05, 54A05, 54A20 Key Words and Phrases: Enriched lattice, L-valued topology, L-valued subgroup, L-valued topological group, Moore collection, Moore closure, L-valued closure group, Kent convergence group, category theory 1. Introduction We have investigated a notion of L-valued topological groups in [3], where we considered L-valued subgroup of a group. Various aspects of L-valued subgroups of groups are studied over the years by various authors, cf. [11, 23, 25, 26, 29] but its categorical behaviors are explored in a certain extent in recent times [26], although the category of fuzzy sets being studied for quite a long time, cf. [14, 33]. In [3], we also considered L-valued closure of an L-valued subgroup of a group in the context of L-valued neighborhood groups, where the lattice under consideration was a complete MV-algebra with square roots. Although our main objective of this paper is to explore further L-valued subgroups from categorical view point and study category of L-valued closure spaces vis-à-vis category of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4021 Email addresses: tmga1@ksu.edu.sa (T.M.G. Ahsanullah), thukair@ksu.edu.sa (Fawzi Al-Thukair) http://www.ejpam.com 949 c© 2021 EJPAM All rights reserved. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 950 L-valued closures groups in conjunction with L-valued topological groups, we add some results on the connection of L-valued topological groups and classical Kent convergence groups. However, we mainly focused on the impact of L-valued closure structures on L- valued topological groups instead of convergence groups. We arrange our work as follows. In Section 2, we give a short survey on L-valued structures that we used in the text. The idea of convergence spaces and their connection to topological spaces is quite old, cf. [4–7, 10, 13, 20, 21, 27, 28]; following the concept of the compatibility of convergence structures with groups structures as proposed by D. C. Kent [20], for the first time, we explore a connection between the categories of L-valued topological groups and Kent convergence groups, this is done in Section 3. We introduce the concept of L-valued closure space, and L-closure of L-valued subgroup of a group in Section 4; we also introduce here a category of L-valued closure groups - a topological category. With the help of connections, as presented by L. N. Stout in [32] and C. L. Waker in [33] between the categories of L-SET and L-TOL, the category of L-valued tolerance spaces [32], we prove a connection between L-GRP, category of L-valued subgroups, and L-valued transitive tolerance spaces, L-TranTOL. Section 5 is devoted to study properties of L-valued closure of L-valued subgroups in the context of L-valued topological groups, where some properties from groups are taken into consideration. 2. Preliminaries Throughout the text we consider L = (L,≤) a complete lattice with >, the top element and ⊥, the bottom element of L. Definition 1. [16, 17] A triple (L,≤, ∗), where ∗ : L× L −→ L is a binary operation on L, is called a GL-monoid if and only if the following holds: (GLM1) (L, ∗) is a commutative semigroup; (GLM2) ∀α ∈ L: α ∗ > = α, (GLM3) ∗ is distributive over arbitrary joins: γ ∗ (∨ k∈K αk ) = ∨ k∈K(γ ∗ αk), for k ∈ K, αk, γ ∈ L; (GLM4) for every γ ≤ α there exists β ∈ L such that γ = α ∗ β (divisibility). The triple (L,≤, ∗) is called a commutative quantale if (GLM1)-(GLM3) are fulfilled. If ∗ = ∧, then the triple (L,≤,∧) is called a frame or a complete Heyting algebra. For a commutative quantale, the implication operator→, also known as residuum, is given by → : L× L −→ L, α→ β = ∨ {γ ∈ L|α ∗ γ ≤ β}. A GL-monoid (L,≤, ∗) is called a complete MV-algebra if ∀α ∈ L, (α→ ⊥)→ ⊥ = α (double negation). This means, in particular, that the unary operation ¬ : L −→ L, α 7→ ¬α = α → ⊥ is an order-reversing involution. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 951 Definition 2. [16, 17] A triple (L,≤,⊗), where ⊗ : L×L −→ L is a binary operation on L, is called a co-premonoid if and only if the following conditions are fulfilled: (CP1) ∀α1, α2, β1, β2 ∈ L: α1 ≤ β1 and α2 ≤ β2 implies α1 ⊗ α2 ≤ β1 ⊗ β2; (CP2) ∀α ∈ L: α ≤ α⊗> and α ≤ >⊗ α. The category COPML consists of all co-premonoids as objects and morphisms as the mappings ι : (L1,≤1,⊗1) −→ (L2,≤2,⊗2) satisfying the following conditions: (CPM1) ι preserves arbitrary joins; (CPM2) ι (α⊗1 α ′) = ι(α)⊗2 ι(α ′), ∀α, α′ ∈ L1; (CPM3) ι preserves universal upper bounds; i.e., ι(>) = >. Definition 3. [16, 17] A co-premonoid (L,≤,⊗) is called a cl-premonoid if and only if (CP3) γ ⊗ (∨ k∈K αk ) = ∨ k∈K(γ ⊗ αk), and (∨ k∈K αk ) ⊗ γ = ∨ k∈K (αk ⊗ γ) for K 6= ∅, k ∈ K, αk, γ ∈ L, is satisfied. Definition 4. [16, 17] The quadruple (L,≤, ∗,⊗) is called an enriched cl-premonoid if and only if the following are fulfilled: (CLP1) (L,≤, ∗) is a GL-monoid; (CLP2) (L,≤,⊗) is a cl-premonoid; (CLP3) ∗ is dominated by ⊗: ∀α, β, γ, δ ∈ L, (α⊗ β) ∗ (γ ⊗ δ) ≤ (α ∗ γ)⊗ (β ∗ δ). Definition 5. [16, 17] A GL-monoid (L,≤, ∗) is said to have square roots if and only if there exists a unary operator S : L −→ L such that the conditions below are satisfied: (S1) S(α) ∗ S(α) = α, ∀α ∈ L; (S2) β ∗ β ≤ α implies β ≤ S(α). Since the formation of square roots is uniquely determined by (S1) and (S2), S(α) is also written as α 1 2 . A GL-monoid with square roots satisfies (S3) if it fulfills the following axiom: (S3) (α ∗ β) 1 2 = ( α 1 2 ∗ β 1 2 ) ∨ ⊥ 1 2 , ∀α, β ∈ L. If L = (L,≤, ∗) is a GL-monoid with square roots, then the monoidal mean operator ~ : L× L −→ L is given by α ~ β = α 1 2 ∗ β 1 2 , ∀α, β ∈ L. An enriched cl-premonoid L = (L,≤, ∗,⊗) is said to be pseudo-bisymmetric if it satisfies the following axiom: (α ∗ β) ⊗ (γ ∗ δ) = ((α⊗ γ) ∗ (β ⊗ δ)) ∨ ((α⊗⊥) ∗ (β ⊗>)) ∨ ((⊥⊗ γ) ∗ (>⊗ δ)), ∀α, β, γ, δ ∈ L. Remark 1. [16, 17] (1) If (L,≤, ∗) is a GL-monoid with square roots, satisfying (S3), and ⊗ is the monoidal mean operator ~, then the quadruple (L,≤, ∗,⊗) is pseudo-bisymmetric. (2) If the cl-premonoid operation ⊗ is identical to the quantal operation ∗, that is, ⊗ = ∗, then the triple (L,≤, ∗,⊗) is pseudo-bisymmetric. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 952 Proposition 1. [18] Let (L,≤, ∗) be a GL-monoid. Then the following are fulfilled ∀α, β, γ, δ, αj , βj , γj ∈ L: (1) α ≤ β → γ ⇔ α ∗ β ≤ γ; (2) α ∗ (α→ β) ≤ β; (3) α ≤ β ⇒ α→ γ ≤ β → γ; (4) α ≤ β ⇒ γ → α ≥ γ → β; (5) (α→ β)→ β ≥ α; (6) α ∗ (β → γ) ≤ β → (α ∗ γ); (7) α→ ( ∧ j∈J βj) = ∧ j∈J(α→ βj); (8) ( ∨ j∈J αj)→ β = ∧ j∈J(αj → β); (9) if α, β ∈ L with α ≤ β, then for any γ ∈ L, γ ∗ α ≤ γ ∗ β; (10) ∧ j∈J (αj ∗ γj) ≥ (∧ j∈J αj ) ∗ (∧ j∈J γj ) ; (11) (α→ γ) ∗ (β → δ) ≤ α ∗ β → γ ∗ δ; (12) α ≤ β ⇔ α→ β = >; (13) α→ > = >, > → α = α, and ⊥ → α = >. In what follows, the quadruple L = (L,≤, ∗,⊗) (or simply L) is assumed to be an enriched cl-premonoid, where ∗ is reserved for the GL-monoid operation, ⊗ is for cl-premonoid, unless otherwise specified. The set of all L-sets or L-valued sets and is denoted by LX(= {ν : X −→ L}). If f : X → Y is a function, then f← : LY −→ LX is defined for any µ ∈ LY by f←(µ) = µ ◦ f ; and f→ : LX −→ LY is defined by f→(ν)(y) = ∨ {ν(x)|f(x) = y}, for all ν ∈ LX , y ∈ Y . If · is a binary operation on a set X, then we define the binary operation � on LX as follows. For ν1, ν2 ∈ LX and z ∈ X ν1 � ν2(z) = ∨ {ν1(x) ∗ ν2(y)|x, y ∈ X,x · y = z}; usually, we write xy instead of x · y. If ν1, ν2 ∈ LX , and →, ∗, ⊗ are operations on L as explained before, then these operations are carried over to LX point-wise: (i) (ν1 → ν2)(x) = ν1(x)→ ν2(x); (ii) (ν1 ∗ ν2)(x) = ν1(x) ∗ ν2(x); (iii) (ν1 ⊗ ν2) (x) = ν1(x)⊗ ν2(x), ∀x ∈ X. Definition 6. [17, 18] A map F : LX −→ L is called an L-valued filter on X if and only if the conditions below are satisfied: (LF1) F(>X) = >, F(⊥X) = ⊥ ; (LF2) if ν1, ν2 ∈ LX with ν1 ≤ ν2, then F(ν1) ≤ F(ν2); (LF3) F(ν1)⊗F(ν2) ≤ F(ν1 ⊗ ν2), ∀ν1, ν2 ∈ LX . (SL) An L-valued filter F is called a stratified L-valued filter if ∀α ∈ L,∀µ ∈ LX , α ∗ F(µ) ≤ F(α ∗ µ). The set of all stratified L-valued filters on X is denoted by FsL(X). On FsL(X), partial ordering ≤ is defined by: if F ,G ∈ FsL(X), then F ≤ G ⇔ F(ν) ≤ G(ν), ∀ν ∈ LX . If T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 953 x ∈ X, then [x] ∈ FsL(X), called point stratified L-valued filter on X, and is defined as [x](ν) = ν(x), for all ν ∈ LX . If F ∈ FsL(X), then the stratified L-valued filter f⇒(F) : LY → L on Y is defined for any µ ∈ LY by [f⇒(F)](µ) = F (f←(µ)) = F(µ ◦ f). If F ∈ FsL(Y ), then f⇐(F) : LX → L is defined by [f⇐(F)](ν) = ∨ {F(µ)|µ ∈ LY , f←(µ) ≤ ν}, for all ν ∈ LX , is a stratified L-filter on X if and only if for all µ ∈ LY , f←(µ) = ⊥X ⇒ F(µ) = ⊥. If ν ∈ LX and µ ∈ LY , then the product ν × µ : X × Y −→ L is defined by: ν × µ = ν ◦ pr1 ∗ µ ◦ pr2, where pr1 : X×Y → X, (x, y) 7→ x and pr2 : X×Y → Y, (x, y) 7→ y are usual projections. Note that in the preceding definition of product L-set the operation ∗ holds only for finite case; otherwise, we need to take ∗ = ∧. Proposition 2. [16] If (L,≤, ∗) is a GL-monoid, then for stratified L-valued filters F1 and F2, the supremum F1∨F2 exists if and only if F1(ν1)∗F2(ν2) = ⊥ ∀ν1, ν2 ∈ LX such that ν1 ∗ ν2 = ⊥X . In particular, the supremum is the stratified L-valued filter defined for all ν ∈ LX by F1 ∨ F2(ν) = ∨ {F1(ν1) ∗ F2(ν2)| ν1, ν2 ∈ LX , ν1 ∗ ν2 ≤ ν}. Let (G, ·) be a group. If F ∈ Ls(G), then F−1 is defined by F−1(ν) = F(ν−1), where ν−1 : G −→ L, x 7−→ ν(x−1). Clearly, F−1 ∈ FsL(G), since for any ν ∈ LX , ⇒(F)(ν) = F (←(ν)) = F(ν−1) = F−1(ν), where  : G −→ G, x 7→ x−1. Also, if m : G × G → G, (g, h) 7→ gh, then for any ν1, ν2 ∈ LG and z ∈ G, m→ (ν1 × ν2) (z) = ∨ m(g,h)=z (ν1 × ν2) (g, h) = ∨ gh=z (ν1 ◦ pr1 ∗ ν2 ◦ pr2) (g, h) = ∨ gh=z ν1 ◦ pr1(g, h) ∗ ν2 ◦ pr2(g, h) = ∨ gh=z ν1(g) ∗ ν2(h) = ν1 � ν2(z). Lemma 1. [3] Let L = (L,≤, ∗) be a GL-monoid and (G, ·) ∈ |GRP|. Then for any F ,G ∈ FsL(X), m⇒(F × G) = F � G. Definition 7. [17] Consider a mapping N : X −→ LLX such that the following conditions are fulfilled: (LN1) Nx(>X) = >; (LN2) Nx(ν1) ≤ Nx(ν2) for all ν1, ν2 ∈ LX with ν1 ≤ ν2; (LN3) Nx(ν1)⊗Nx(ν2) ≤ Nx(ν1 ⊗ ν2), for all ν1, ν2 ∈ LX ; (LN4) Nx(ν) ≤ ν(x), for all ν ∈ LX ; (LN5) ∀x ∈ X and ν ∈ LX , Nx(ν) ≤ ∨ {Nx(µ) : µ ∈ LX , µ(y) ≤ [Ny](ν), ∀y ∈ X} (SLN) α ∗Nx(ν) ≤ Nx(α ∗ ν). Then N = (Nx)x∈X is called a stratified L-valued neighborhood system on X, and the T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 954 pair (X,N = (Nx)x∈X) is called a stratified L-valued neighborhood space. If (X,N) and (Y,M) stratified L-valued neighborhood spaces, then a map f : (X,N) → (Y,M) is said to be continuous at a point x ∈ X if and only if Mf(x)(ν) ≤ Nx (f←(ν)), for all ν ∈ LY . SL-NS denotes the category of all stratified L-valued neighborhood spaces as objects and all continuous maps as morphisms. Definition 8. [17, 22] Let ∆ ⊆ LX such that the following are fulfilled: (LT1) >X ,⊥X ∈ ∆; (LT2) ν1, ν2 ∈ ∆⇒ ν1 ⊗ ν2 ∈ ∆; (LT3) {νj}j∈J ⊆ ∆⇒ ∨ j∈J νj ∈ ∆; (SLT) ν ∈ ∆, α ∈ L⇒ αX ∗ ν ∈ ∆. We call ∆ an L-valued topology on X if it satisfies (LT1)-(LT3), and the pair (X,∆) is called an L-valued topological space. If ∆ satisfies (LT1)-(SLT) then we call it a stratified L-valued topology on X and the pair (X,∆) or X in short, if there is no confusion, is called a stratified L-valued topological space; members of ∆ are called open L-valued sets or L-valued subsets; the members of Θ(X) = {ξ ∈ LX : ξc is open} are called closed L- valued sets or L-valued subsets, where ξc is the so-called qusi-complementation of ξ. Note that Θ(X) is closed under formation of arbitrary infs and finite sups. Furthermore, recall that the closure of ν ∈ LX , denoted by νX is defined as: νX = ∧ {θ ∈ Θ(X) : ν ≤ θ}. If (X,∆) and (Y,Γ) are stratified L-valued topological spaces, then a function f : (X,∆)→ (Y,Γ) is said to be continuous if and only if for any σ ∈ Γ, f←(σ) ∈ ∆. The category SL-TOP consists of all stratified L-valued topological spaces as objects and all continuous maps between them as morphisms, while the category L-TOP consisting of all L-valued topological spaces as objects and all continuous maps between them as morphisms. Every stratified L-valued topology ∆ on X induces a stratified L-valued neighborhood system N∆ = (Nx ∆) as follows: Nx ∆(µ) = ∨ {ν(x) : ν ∈ ∆, ν ≤ µ}, for all µ ∈ LX and x ∈ X. Conversely, every stratified L-valued neighborhood system N = (Nx)x∈X on X induces a stratified L-valued topology ∆N on X : ∆N = {ν ∈ LX : ν(x) ≤ Nx(ν), ∀x ∈ X}. It follows that the interrelationship between L-valued neighborhood system and L-valued topologies can be viewed as: ν ∈ ∆⇔ ν(x) ≤ Nx(ν), ∀x ∈ X (†). As a consequence of (†) it follows that the continuity between the objects in SL-TOP, and the continuity between objects in SL-NS are equivalent concept, cf. [18]. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 955 3. L-valued topological groups and Kent convergence groups We consider L = (L,≤, ∗,⊗ = ∗) an enriched cl-premonoid, where ∗ is a GL-monoid operation. Let the category of groups and group homomorphisms be denoted by GRP. Definition 9. Let (X, ·) ∈ |GRP| and (X,∆) ∈ |SL-TOP|. Then the triple (X, ·,∆) is called a stratified L-valued topological group if and only if the conditions below are fulfilled: (LTGM) the mapping m : (X ×X,∆×∆) −→ (X,∆), (x, y) 7−→ xy is continuous ; (LTGI) the mapping  : (X,∆) −→ (X,∆), x 7−→ x−1 is continuous. The category of all stratified L-valued topological groups and continuous group homomor- phisms is denoted by SL-TOPGRP. Definition 10. [3] Let (X, ·) ∈ |GRP| and ( X,N = (Nx)x∈X ) ∈ |SL-NS|. Then the triple ( X, ·,N = (Nx)x∈X ) is called a stratified L-valued neighborhood group if and only if (LNGM) Nxy ≤ Nx�Ny, and (LNGI) Nx−1 ≤ (Nx)−1 are satisfied, where for any ξ ∈ LG: Nx �Ny(ξ) = m⇒ (Nx ×Ny) (ξ) = ∨ {Nx(ξ1) ∧Ny(ξ2) : ξ1, ξ2 ∈ LX , ξ1 × ξ2 ≤ m←(ξ)}. A stratified L-valued neighborhood system on a group X is said to be compatible with the group structure of X if and only if the group operations are continuous; i.e., conditions (LNGM) and (LNTGI) are fulfilled. The category SL-NS consists of all stratified L-valued neighborhood groups as objects and continuous group homomorphisms as morphisms. Example 1. Let (G, ·) ∈ |GRP|, and Ri : LX −→ L defined by Ni = ∧ x∈G[x]. Then the triple (G, ·,Ni) is a stratified L-valued neighborhood group, called indiscrete stratified L-valued neighborhood group. Example 2. Let (G, ·) ∈ |GRP|, and Rd : LX −→ L defined by Nxd(ν) = ν(x). Then the triple (G, ·,Nd) is a stratified L-valued neighborhood group, called discrete stratified L-valued neighborhood group. Lemma 2. [3] Let (G, ·,∆) ∈ |SL-TOPGRP|, and a ∈ G. Then the translations (left and right) La : (G, ·,∆) −→ (G, ·,∆) , g 7−→ ag, and Lx : (G, ·,∆) −→ (G, ·,∆) , g 7−→ ga are homeomorphisms. Also the mapping Ca : (G, ·,∆) −→ (G, ·,∆) , g 7−→ gag−1 the inner automorphism is an isomorphism. Definition 11. [20, 27] A Kent convergence structure q on X is a subset q ⊆ F(X)×X such that the following conditions are satisfied: (C1) x ∈ q(ẋ),∀x ∈ X, where ẋ denotes the ordinary principal filter on X generated by the singleton {x}; (C2) F,G ∈ F(X), F ⊆ G, x ∈ q(F) implies x ∈ q(G); (C3) x ∈ q(F) implies x ∈ q(F ∩ ẋ). Note that in [4], [6] and [7] the above notion is called a local filter convergence structure q on X, however. A mapping f : (X, q) −→ (X ′, q′) is called continuous if for all F ∈ F(X) and x ∈ X, x ∈ T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 956 q(F) implies f(x) ∈ q(f(F)). The category of all Kent convergence spaces and continuous mapping is denoted by KCONV. The category KCONV is a strong topological universe, cf. [10, 28]. The pair (X, q) is called a limit space if conditions (C1), (C2) and (C4): ∀F,G ∈ F(X), x ∈ q(F) and x ∈ q(G) implies x ∈ q(F ∩G). The category of limit spaces is denoted by LIM. A limit structure q on X is called a principal limit structure on X if and only if for every x ∈ X there exists a unique filter Ux ∈ F(X) such that the following relation holds: q = {(F, x) ∈ F(X)×X : Ux ⊆ F}. The category of all principal limit spaces and continuous mappings is denoted by pLIM. Remark 2. It is important to mention here that the categories of closure spaces, CLS, and LIM with principal limit structures are isomorphic, cf. [28], we are not interested at this stage to carry out research in this direction, and postpone it for further investigation. Definition 12. [27] Let (G, ·) ∈ |GRP| and (G, q) ∈ |KCONV| (resp. (G, q) ∈ |LIM|). Then the triple (G, ·, q) ∈ |KCONVGRP| (resp. (G, ·, q) ∈ |LIMGRP|) if the following are fulfilled: (CGM) x ∈ q(F) and y ∈ q(G) implies xy ∈ q (F�G); (CGI) x ∈ q(F) implies x−1 ∈ q(F−1). The category of all Kent convergence groups and group homomorphisms is denoted by KCONVGRP (resp. the category of all limit groups and group homomorphisms is de- noted by LIMGRP). Given a stratified L-topological space (X,∆N) with the corresponding L-neighborhood system N. Then a filter F is said to be convergent to a point x ∈ X (we denoted it as x ∈ q∆N (F)) with respect to ∆N if and only if for all ν ∈ LX the following holds: Nx(ν) ≤ ∨ F∈F (∧ y∈F ν(y) ) . Lemma 3. Let (G, ·,∆N) ∈ |SL-TOPGRP|, where ∆ is a stratified L-valued topol- ogy on G and N is a corresponding L-valued neighborhood system. Then (G, ·, q∆N ) ∈ |KCONVGRP|. Proof. Let (G, ·,∆N) ∈ |SL-TOPGRP|. Then in view of the Lemma 5.4.1[18], we only need to Check the conditions (CGM) and (CGI). (CGM) Let for F,G ∈ F(G) and x, y ∈ G, x ∈ q∆N (F) and y ∈ q∆N (G). Then for any ν, µ ∈ LG: Nx(ν) ≤ ∨ F∈F ∧ y1∈F ν(y1), and Ny(µ) ≤ ∨ G∈G ∧ y2∈G µ(y2). Thus, for any σ ∈ LG, Nxy(σ) ≤ ∨ {Nx(ν) ∗Ny(µ) : ν(x) ∗ µ(y) ≤ σ(xy)} ≤∨ ν(x)∗µ(y)≤σ(xy) ∨ F ·G∈F�G ∧ y1∈F,y2∈G ν(x) ∗ µ(y) ≤ ∨ F ·G∈F�G ∧ xy∈F ·G σ(xy) T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 957 This implies that Nxy(σ) ≤ ∨ F ·G∈F�G ∧ z∈F ·G σ(xy), i.e., xy ∈ q∆N (F�G). (CGI) Let F ∈ F(G), and x ∈ X. Then by invoking (†) in conjunction with the Lemma 5.4.1[18], if we consider x ∈ q∆N (F), then for any ν ∈ LG, we have Nx(ν) ≤ ∨ F∈F (∧ y∈F ν(y) ) . Now due to the continuity of , we have Nx−1(ν) ≤ Nx(ν−1) ≤ ∨ F∈F (∧ y∈F ν −1(y) ) = ∨ F−1∈F−1 (∧ y−1∈F−1 ν(y−1) ) . That is, Nx−1(ν) ≤ ∨ F−1∈F−1 (∧ y−1∈F−1 ν(y−1) ) implying x−1 ∈ q∆N (F−1). Remark 3. Referring to the pp. 175 [18], one can observe that given a Kent convergence structure q on X, then q induces a stratified L-valued topology ∆̂q in the following way: ∆̂q = {σ ∈ LX : σ(x) ≤ ∨ A∈F (∧ z∈A σ(z) ) , ∀F ∈ F(X), x ∈ q(F)} From Lemma 5.4.2[18], it follows that there is a functor G : KCONV −→ SL-TOP, where G(X, q) = (X, ∆̂q) and G(f) = f . Lemma 4. Let (G, ·, q) ∈ |KCONVGRP|. Then (G, ·, ∆̂q) ∈ |SL-TOPGRP|. Proof. Let (G, ·, q) ∈ |KCONVGRP|. Note that the product L-valued topology on ∆̂q × ∆̂q is the initial L-valued topology with respect to the projects pr1 : X × X −→ X,(x, y) 7−→ x, and pr2 : X × X −→ X,(x, y) 7−→ y. Further note that ∆̂q × ∆̂q = {(ν1 · pr1) ∗ (ν2 · pr2) : ν1, ν2 ∈ ∆̂q} is a base for the product L-topology on X ×X, where the L-set can be given by: µ0 := ∨ i∈I ( ν1 i · pr1) ∗ (µ2 i · pr2) ) , and ν1 i , µ 2 i ∈ ∆̂q. Thus, we have for any ν ∈ ∆̂q and (x, y) ∈ X ×X, and due to the property of ∗ in L: ν(xy) = m←(ν)(x, y) = ∨ i∈I [( pr←1 (ν1 i )(x, y)) ∗ (pr←2 (µ2 i )(x, y)) )] (ν1 i , µ 2 i ∈ ∆̂q). = ∨ i∈I [ ν1 i (x) ∗ µ2 i (y) ] , (ν1 i , µ 2 i ∈ ∆̂q). ≤ ∨ i∈I [∨ A∈F (∧ z1∈A ν 1 i (z1) ) ∗ ∨ B∈G (∧ z2∈B ν 2 i (z2) )] ≤ ∨ i∈I [∨ A.·B∈F�G ∧ z1z2∈A·B ( ν1 i (z1) ∗ ν2 i (z2) )] = [∨ A·B∈F�G ∧ z1z2∈A·B ν(z1z2) ] That is, ν(xy) ≤ [∨ H∈F�G ∧ z1z2∈H ν(z1z2) ] and xy ∈ q (F�G) due to the condition (CGM) implying m←(ν) ∈ ∆̂q × ∆̂q. This proves condition (LTGM). Now let x ∈ q(F) for any F ∈ F(G) and let ν ∈ ∆̂q. Then we have ←(ν)(x) = ν((x)) ≤ ∨ A∈F (∧ z2∈(A) ν(z2) ) = ∨ A−1∈F−1 (∧ z1∈A−1 ←(ν)(z1) ) , that is, ←(ν)(x) ≤ ∨ A−1∈F−1 (∧ z1∈A−1 ←(ν)(z1) ) ; and x−1 ∈ q(F−1) because of the condition (CGI). These together imply that ←(ν) ∈ ∆̂q, this proves (LTGI). Theorem 1. The functor F : SL-TOPGRP−→ KCONVGRP as defined below F :  SL-TOPGRP −→ KCONVGRP (G, ·,∆N) 7−→ (G, ·, q∆N ) f 7−→ f has a left adjoint. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 958 Proof. In view of Lemma 3 in conjunction with Lemma 5.4.1 [18], F : SL-TOPGRP−→ KCONVGRP is a functor. Define G : KCONVGRP −→ SL-TOPGRP by G :  KCONVGRP −→ SL-TOPGRP (G, ·, q) 7−→ ( G, ·, ∆̂q ) f 7−→ f Then from Lemma 4 in conjunction with Lemma 5.4.2 [18] that G is a functor since in both the cases the group homomorphism structures remain unchanged. That the functor G is a left adjoint since in both the cases group homomorphism structures remain unchanged. That the functor G is a left adjoint to F is an immediate consequence of the Proposition 5.4.3 [18]. 4. Enriched lattice-valued subgroup of a group and enriched lattice-valued neighborhood groups Definition 13. Let L = (L,≤,∧, ∗) be an enriched cl-premonoid, (G, ·) ∈ |GRP|. Then an L-set µ : G −→ L is called an L-valued subgroup of a group G if and only if the following conditions are fulfilled: (LG1) µ(e) = >; (LG2) µ(g) ∗ µ(h) ≤ µ(gh), ∀g, h ∈ G; (LG3) µ(g) ≤ µ(g−1). Then the pair (G, ·, µ) is called an L-valued subgroup space. Let (H, ·, ξ) be another L-valued subgroup of a group H. Define a mapping between L-valued subgroup spaces, f : (G, ·, µ) −→ (H, ·, ξ) such that µ(g) ≤ ξ(f(g)), ∀g ∈ G (‡) The category of all L-valued subgroup spaces and all group homomorphisms satisfying (‡) is denoted by L-GRP. Sometime we denote the set of L-valued subgroups of a group G by L(G). Example 3. [3] Let L = ([0, 1],≤,∧, ∗) be an enriched cl-premonoid, where ∗ is a t- norm on [0, 1]. Let G be the cyclic group Cn of order n (n ≥ 1) with a as the generator; specifically, Cn = {e, a, a2, ..., an−1; an = e} with respect to multiplication ·. Define µ : G→ [0, 1] by µ(x) = { 1, if x = e; 1 n , otherwise. Then (G, ·, µ) is an enriched lattice-valued subgroup space. In fact, for (LG1) µ(e) = 1 while (LG3) follows from the definition. For (LG2), consider x, y ∈ G with x 6= e and y 6= e, then µ(x) ∗ µ(y) = 1 n ∗ 1 n ≤ 1 n ∗ 1 = 1 n implying µ(x) ∗ µ(y) ≤ µ(xy); other choices follow similarly. Hence µ is an enriched lattice-valued subgroup of the group G. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 959 Remark 4. In [33], C. L. Walker pointed out that for a category of fuzzy subsets F = Set(I) where all objects are (X, ν), X ∈ |Set|, with ν : X −→ I - a mapping from X to the unit interval. The morphisms F are all mappings f : (X, ν) −→ (Y, µ) satisfying ν(x) ≤ µ(f(x)). Furthermore, note that in [14], J. Goguen, defined the category SET(L) having objects the pair (X, ν), where ν : X −→ L, and morphisms f : (X, ν) −→ (Y, µ) such that ν(x) ≤ µ(f(x)) holds. L. Stout [32] argued that this category SET(L) has initial structure and is cartesian closed. The initial structure is given as: for a family of mappings (fj : X −→ (Yj , µj))j, ν(x) = ∧ j µj(fj(x)) gives the initial structure on X. The cartesian closed structure is obtained as: (C(X,Y ),∇), where ∇(f) = ∧ x∈X [ν(x) −→ µ(f(x))], where for all (f : (X, ν) −→ (Y, µ)) ∈ C(X,Y ), and the implication → is given by: ν(x) −→ µ(f(x)) = ∨ {λ : λ ∧ ν(x) ≤ µ(f(x))}. Lemma 5. L-GRP has initial structure where the underlying forgetful functor is given by T : L-GRP−→ GRP. Proof. Consider a group (G, ·) and a family of mappings (fj : G −→ (Hj , µj))j∈J , where each fj : G −→ Hj is a group homomorphism, µj is a subgroup of Hj , for each j ∈ J . Then the structure on µ on G is given by ν(g) = ∧ j µj(fj(g))(= ∧ j f ← j (µj)(g)), for all g ∈ G, note that for each j ∈ J , f←j (µj) is also an L-subgroup of G, and the arbitrary intersection ν is also an L-subgroup of G, and hence (G, ·, ν) ∈ |L-GRP|. Let (Z, ·, %) ∈ |L-GRP|, we prove that the mapping ϕ : (Z, ·, %) −→ (G, ·, ν) a group homomorphism is an L-GRP- morphism if and only if fj ◦ ϕ : (Z, ·, %) −→ (Hj , ·, µj) is an L-GRP-morphism. We only show g : (Z, ·, %) −→ (G, ·, ν) is an L-GRP-morphism. So, for any z ∈ Z, %(z) ≤ µj(fj(ϕ(z)) = ∧ j∈J f ← j (µj)(ϕ(z)) = ν(ϕ(z)), i.e., %(z) ≤ ν(ϕ(z)). Theorem 2. Let L = (L,≤, ∗ = ∧) be a complete Heyting algebra, and (G, ·, ν) be an L-valued subgroup space and T (G) = {f : (G, ν) −→ (G, ν); f is bijective and both f and f−1 satisfy (‡) }. Then (T (G), ·,∇) is an L-subgroup space, where (fg)(x) = f(x)g(x) and f−1(x) = (f(x))−1. Proof. Clearly (T (G), ·) is a group under composition. Define ∇(f) = ∧ x∈G[ν(x)→ ν(f(x))], ∀f ∈ T (G) (a) and ∇(−1)(f) = ∧ x∈G[ν(x)→ ν(f−1(x))], ∀f ∈ T (G) (b) Combining (a) and (b) it follows upon using Proposition 1(7) that ∇(f) = ∧ x∈G[ν(x)→ ν(f(x))∧ ν(f−1(x))]. Then clearly (LG1) and (LG3) are true upon using Proposition 1(7) and (LG2), i.e., ∇(idG) = >, and ∇(f) ≤ ∇(f−1); we only look at (LG2). For, let f, g ∈ T (X), then we have ∇(f) ∧∇(g) = ∧ x∈G[ν(x)→ ν(f(x)) ∧ ν(f−1(x))] ∧ ∧ x∈G[ν(x)→ ν(g(x)) ∧ ν(g−1(x))] ≤ ∧ x∈G[ν(x)→ ν(f(x))∧ ν(g(x))∧ ν(g−1(x))∧ ν(f−1(x))] ≤ ∧ x∈G[ν(x)→ ν(f(x)g(x))∧ ν(g−1(x)f−1(x))] = ∧ x∈G[ν(x)→ ν(fg(x)) ∧ ν((fg)−1(x))] = ∇(fg). T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 960 Definition 14. [23, 25] An L-valued subgroup is called L-valued normal subgroup if for all x, y ∈ G if it satisfies one of the following equivalent conditions: (1) ν(xy) = ν(yx); (2) ν(xyx−1) ≥ ν(y); (3) ν(xyx−1) = ν(y). Definition 15. A mapping ` : LX −→ LX is said to be an L-valuedclosure operation on X if the following conditions hold for every ν, µ ∈ LX : (1) ν ≤ `(ν); (2) ν ≤ µ implies `(ν) ≤ `(µ); (3) `(`(ν)) = `(ν); (4) `(>∅) = ⊥. The pair (X, `) is called is called an L-valued closure space and ν ∈ LX is called closed if ν = `(ν). Note that (2) implies `(ν) ∨ `(µ) ≤ `(ν ∨ µ), for any ν, µ ∈ LX . The category of all L-valued closure spaces and all closure preserving mappings, i.e., map- pings f : (X, `) −→ (Y, `) that satisfy f→(`(ν)) ≤ ` (f→(ν)) for all ν ∈ LX , is denoted by L-CLS. Lemma 6. We have the following forgetful functor forgetting L-valued closure structure: U :  L-CLS −→ SET(L) (X, `) 7−→ (X, ν) f 7−→ f where U((X, `)) = (X, ν) and for f : X −→ Y , U(f) = f , f→ : LX −→ LY , and U(f) yields an SET(L)-morphism. Let X ∈ |SET| and let Ω ⊂ LX be a collection of L-subsets of X. Then we call Ω a lattice- valued Moore collection if every intersection of members of Ω belongs to Ω, i.e., given a family (νj)j∈J of L-subsets: ∀j ∈ J , νj ∈ Ω =⇒ ∧ j∈J νj ∈ Ω. If Ω is a lattice-valued Moore collection containing >∅, then if `(µ)Ω = ∧ {ν ∈ Ω: µ ≤ ν, ν is L-valued closed set}, i.e. if `(µ) is the intersection of all L-valued closed sets that contain µ, then ` is an L-valued closure operator. We refer to Birkhoff [9], and Schechter [31], for the classical notion of Moore collection. Example 4. L-valued subgroups of a group (G, ·) form a lattice-valued Moore collection; this is so, since arbitrary intersection of L-valued subgroups is again an L-valued subgroup, cf. [11], pp. 115. In fact, if we let µ = ∧ j∈J νj, then we can easily verify the Definition 13. In fact, (LG1) µ(e) = ∧ νj(e) = > for all j ∈ J ; (LG2) upon using Proposition 1(10), we have: µ(x)∗µ(y) = ( ∧ j∈J νj(x))∗( ∧ j∈J νj(y)) ≤ ∧ j∈J (νj(x) ∗ νj(y)) ≤ ∧ j∈J νj(xy) = µ(xy), so, µ(x) ∗µ(y) ≤ µ(xy); (LG3) µ = ∧ j∈J(νj(x)) ≤ ∧ j∈J(νj(x −1)) = µ(x−1). Also, if µ ∈ LG, then L-valued closure of µ is the subgroup generated by µ. This can be given as: 〈`(µ)〉 = ∧ {ν : µ ≤ ν, ν is closed LG-valued subgroup of G}, T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 961 the L-valued subgroup that contains µ. In view of the Theorem 5.2.6[11], normal L-valued subgroup of the group G form a lattice- valued Moore collection, and in particular, `(µ), µ ∈ LG is the normal L-valued subgroup generated by µ. More precisely, 〈`(µ)〉 = ∧ {ν : µ ≤ ν, ν is closed normal LG-valued subgroup of G}, Theorem 3. L-CLS is a topological category. Proof. Note that the objects of L-CLS are structured sets and the composition of closure preserving mappings is closure preserving. Consider X is a set, ( Yj , ` j ) j∈J a family of L-valued closure spaces and a source S =( fj : X −→ (Yj , ` j) ) j∈J of family of functions, then Ω = {ω ∈ LX : ω = ∧ j∈J f ← j (ωj), ∀ωj = `j(ωj), j ∈ J} is a lattice-valued Moore family which contains >∅. Then Ω induces an L-valued closure operation on X given by: `(µ)Ω = ∧ {ω ∈ Ω: µ ≤ ω}, for all µ ∈ LX . Now let (Z, `) ∈ |L-CLS|, and g : Z −→ X be a function such that fj ◦ g : (Z, `) −→ (Y, `j) is closure preserving mapping for all j ∈ J . If µ ∈ LX is a − Ω closed, then µ ∈ Ω and thus µ = ∧ j∈J f ← j (ωj) where ωj = `j(ωj) in (Yj , `j). In view of Proposition 1.2(5) [22], we have: g←(µ) = g← ∧ j f←j (ωj)  = ∧ j g← ( f←j (ωj) ) = ∧ j (fj ◦ g)←(ωj) This implies (fj ◦ g)←(ωj) is closed in (Z, `) implying g←(µ) is closed in (Z, `). Remark 5. Every L-valued topological space (X,∆) is an L-valued closure space with the closure operation defined by: `(ν) = ν(X,∆) = νX for every ν ∈ LX . Also, every mapping f : (X,∆) −→ (Y,Γ) continuous if and only if it is closure preserving with respect to the induced L-valued closure operations. In fact, if ν ∈ LX , then in view of the Proposition 1.4 [22], f→(`(ν)) = f→(νX) ≤ f→(ν) Y = `(f→(ν)), i.e., f→(`(ν)) ≤ `(f→(ν)), mean- ing f is closure preserving. Conversely, let ν ∈ LX and f be closure preserving, then f→(νX) = f→(`(ν)) ≤ `(f→(ν)) = f→(ν) Y , i.e., f→(νX) ≤ f→(ν) Y meaning the map- ping f : (X,∆) −→ (Y,Γ) is continuous by the Proposition 1.4 [22]. Thus we have the following. Corollary 1. L-TOP, the category of L-valued topological spaces and continuous map- pings is a full subcategory of the category L-CLS Definition 16. A triple (G, ·, `) is called an L-closure group if (G, ·) ∈ |GRP| and (G, `) ∈ |L-CLS| such the following are fulfilled: (clGM) `(ν)(x) ∗ `(ν)(y) ≤ `(ν · ν)(xy), ∀ν ∈ LG and ∀x, y ∈ G; (clGI) `(ν)(x) ≤ `(ν−1)(x−1), ∀ν ∈ LG and x ∈ G. The category of all L-valued closure groups and closure-preserving group homomorphisms is denoted by L-CLGRP. T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 962 Remark 6. If we consider each ν ∈ L(G), i.e., each ν ∈ LG is an L-valued subgroup of the group G, then we obtain a category L-CLGRP∗ of all L-valued closure of L-valued subgroups of G, and closure-preserving mappings. Then L-CLGRP∗ is a subcategory of L-CLGRP. Theorem 4. L-CLGRP is a topological category. Proof. Consider (G, ·) a group, and a source S = (fj : (G, ·) −→ (Gj , ·, `j))j∈J of family of functions, where for each j ∈ J , fj : G −→ Gj is a group homomorphism, then Ω = {ω ∈ LG : ω = ∧ j∈J f ← j (ωj), ∀ωj = `j(ωj), j ∈ J} In view of Theorem 3, we have (G, ·, `) is an L-valued closure space. We only verify (clGM). So we have: `(ω)(x) ∗ `(ω)(y) = ∧ j∈J f ← j (ωj)(x) ∗ ∧ j∈J f ← j (ωj)(y) ≤ ∧ j∈J f ← j (ωj)� f←j (ωj)(xy) = ∧ j∈J f ← j (ωj � ωj)(xy) ≤ `(ω · ω)(xy). Definition 17. [8, 19, 32] An L-tolerance space is a pair (X, τ), where τ : X ×X −→ L such that (T1) τ(x, x) = >, ∀x ∈ X (reflexivity); (T2) τ(x, y) = τ(y, x) (symmetry). If, in addition τ satisfies (T3) τ(x, y) ∗ τ(y, z) ≤ τ(x, z), for any x, y, z ∈ X, then we speak of transitive tolerance relation which is essentially gives an L-equivalence relation. A mapping between L-valued tolerance spaces (resp. transitive L-valued tolerance spaces): f : (X, τ) −→ (Y, τ ′) is called L-valued tolerance preserving if τ(x, y) ≤ τ ′(f(x), f(y)). The category of all L-valued tolerance spaces and L-tolerance preserving mappings is de- noted by L-TOL while L-TranTOL denotes the category of transitive L-tolerance spaces. For an MV-valued algebra L, given L-TranTOL a category of transitive L-valued toler- ance spaces and L-valued tolerance preserving mappings, one can obtain a functor A : L- TOL−→ L-SET where A(X, τ) = (X, τD), D : X −→ X ×X and A(f) = f , here A(f) sends f to an L-tolerance preserving mapping to f : (X, τD) −→ (Y, τ ′D), i.e., τD(x) = τ(x, x) ≤ τ ′(f(x), f(x)) = τ ′D(f(x)), i.e., τD(x) ≤ τ ′D(f(x)). Conversely, given L-SET, one obtains a functor B : L-SET−→ L-TranTOL as defined by: B(X, ν) = (X, τ := ν∧ν) and B(f) = f , τ(x, y) = ν(x) ∧ ν(y) ≤ ν(f(x)) ∧ ν(f(y)) = τ(f(x), f(y)). In view of [11], pp 148, for a group (G, ·), we consider a mapping %L : LG −→ LG×G defined by: %L(ν)(x, y) = ν(x−1y), and analogously, %R(ν)(x, y) = ν(xy−1). Then we have the following. Lemma 7. Let (G, ·) ∈ |GRP|, and the category L-TranTOL consists of morphisms f : (G, τ) −→ (H, %′) which are L-valued tolerance preserving such that each morphism is a group homomorphism. Then A :  L-GRP −→ L−TranTOL (G, ν) 7−→ (G, %L(ν)) f 7−→ f T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 963 Proof. Let ν ∈ L(G), then we have ρL(ν)(x, x) = ν(x−1x) = ν(e) = > which is (T1); for (T2), we apply Theorem 5.1.1(5)[11](see also, Theorem 1.2.2[24]) to get ρL(ν)(x, y) = ν(x−1y) = ν((x−1y)−1) = ν(y−1x) = ρL(y, x). Now for any x, y, z ∈ X, ρL(ν)(x, y) ∗ ρL(y, z) = ν(x−1y) ∗ ν(y−1z) ≤ ν(x−1yy−1z) = ν(x−1z) = ρL(ν)(x, z), which is (T3). To check the morphism part, we have for any x, y ∈ G and ν ∈ L(G): τ(x, y) = ρL(ν)(x, y) = ν(x−1y) ≤ ν ′(f(x−1y)) = ν ′((f(x))−1f(y)) = ρL(ν)(f(x), f(y)) = τ ′(f(x), f(y)), i.e., τ(x, y) ≤ τ ′(ν ′)(f(x), f(y)). Lemma 8. Let (G, ·) ∈ |GRP|, and the category L-TranTOL consists of morphisms f : (G, ρL(ν)) −→ (H, ρL(ν ′)) which are L-valued tolerance preserving such that each mor- phism is a group homomorphism. Then B :  L−TranTOL −→ L-GRP (G, %L(ν)) 7−→ (G, ν) f 7−→ f Proof. Let ν ∈ LG, and (G, ρL(ν)) ∈ |L-TranTOL|, it suffices to show that ν ∈ L(G). Thus, for any x ∈ X, ν(e) = ν(x−1x) = ρL(ν)(x, x) = > which is (LG1). For (LG2) is obviously true while for (LG3), we have for any x, y ∈ G: ν(x) ∗ ν(y) = ν(xe) ∗ ν(ey) = ρL(ν)(x, e) ∗ ρL(ν)(e, y) ≤ ρL(ν)(x, y) = ν(x−1y), i.e., ν(x) ∗ ν(y) ≤ ν(x−1y), this happens when we combine (LG2) and (LG3), cf. Theorem 5.1.3[11]. This shows that ν ∈ L(G). For the morphism part, let x ∈ G and ν ∈ LG. Then ν(x) = ν(ex) = ρL(ν)(e, x) ≤ ρL(ν ′)(f(e), f(x)) = ν ′ ( (f(e))−1f(x) ) = ν ′(f(ex)) = ν ′(f(x)), i.e., ν(x) ≤ ν ′(f(x)). 5. Enriched latticed-valued subgroups on lattice-valued neighborhood groups Let L = (L,≤, ∗) be a complete MV-valued algebra with square roots. If ( X,N = (Nx)x∈X ) is a stratified L-neighborhood space, then in view of [12] (page 13), and [18] (page 226), one can see that N induces a closure operator − : LX −→ LX given for any x ∈ X and ν ∈ LX by ν(x) = ([Nx](ν → ⊥))→ ⊥. Theorem 5. [3, 12, 18] (a) Let ( X,N = (Nx)x∈X ) ∈ |SL-NS|. Then ν(x) = ∨ {F(ν) : F ∈ FsL(X), F ≥ Nx}, ∀ν ∈ LX , and ∀x ∈ X. (b) Let ( G, ·,N = (Nx)x∈G ) ∈ |SL-NGRP| and ν ∈ LG be an L-valued subgroup of a group G. Then the L-valued closure ν of ν in (a) is an L-valued subgroup of G. (c) Let (G, ·,N) −→ (H, ·,M) be continuous group homomorphism. Then ν(x) ≤ f→(ν)(f(x)) for all ν ∈ LG and x ∈ G. Moreover, if ν ∈ LG is an L-valued subgroup of G, then f→(ν) is an L-valued subgroup of H, and if µ ∈ LG is a L-valued subgroup of H, then f←(µ) is an L-valued subgroup of G. (d) If ν ∈ LG is an L-valued normal subgroup of a group G, then ν is also an L-valued T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 964 normal subgroup of G. (e) If (G, ·,N) −→ (H, ·,M) is a continuous group homomorphism and µ ∈ LH is an L-valued subgroup of H, then f←(µ) is an L-valued subgroup of G. Proof. (b) follows from the Theorem 5.1[3]. (c) Let ν ∈ LG, and x ∈ G. Then since ν ≤ f←(f→(ν)) due to Definition 6 (LF2), F(ν) ≤ F (f←(f→(ν))) = f⇒(F)(f→(ν)), and since Mf(x) ≤ f⇒(Nx) due to continuity of f , we have ν(x) = ∨ {F(ν) : F ∈ FsL(X),F ≥ Nx} ≤ ∨ {f⇒(F)(f→(ν)) : f⇒(F) ∈ FsL(Y ), f⇒(F)(f→(ν)) ≥ f⇒(Nx)(f→(ν))} ≤ ∨ {f⇒(F)(f→(ν)) : f⇒(F) ∈ FsL(H), f⇒(F)(f→(ν)) ≥Mf(x)(f →(ν))} = ∨ {G(f→(ν)) : G ∈ FsL(Y ), G ≥Mf(x)} = f→(ν)(f(x)), i.e., ν(x) ≤ f→(ν)(f(x)). (d) Let ν ∈ LG be an L-valued normal subgroup of a group G, and consider the mapping Ca : G −→ G defined by Ca(g) = a−1ga; need to that ν is also an L-normal subgroup of G. Note that ν is L-normal subgroup of G if and only if ν(aga−1) = ν(g). Now since the map- ping Ca is continuous, we have ν(g) ≤ Ca(ν)(Ca(g)) = ∨ x∈C←a (g) ν(x) = ∨ Ca(x)=g ν(x) = ν(aga−1), i.e., ν(aga−1) ≥ ν(g), meaning that ν is a normal L-valued subgroup of G. (e) Let (G, ·,N) −→ (H, ·,M) be a continuous group homomorphism, and µ ∈ L(H). Then f←(µ)(e) = ∨ {F(f←(µ)) : F ∈ FsL(G),F ≥ Ne} ≥ ∨ {[e](f←(µ) : [e] ∈ FsL(G), [e] ≥ Ne} ≥ ∨ {µ(e) : [e] ∈ FsL(G), [e] ≥ Ne} = >, whence µ(e) = >, since µ ∈ L(G), implying f←(µ)(e) = >. Now let x, y ∈ G and µ ∈ LH . Then in view of the Definition 1(GLM3), we have: f←(µ)(x)∗f←(µ)(y) = ∨ {F(f←(µ)) : F ∈ FsL(G),F ≥ Nx}∗ ∨ {G(f←(µ)) : G ∈ FsL(G),G ≥ Ny} = ∨ {F(f←(µ)) ∗ G(f←(µ)) : F ,G ∈ FsL(G),F ≥ Nx,G ≥ Ny} ≤ ∨ {F �G(f←(µ)) : F �G ∈ FsL(G),F �G ≥ Nx�Ny} (By applying Theorem 1.2.8 and Theorem 1.2.11[24], whence f←(µ) ∈ L(G)) ≤ ∨ {F � G(f←(µ)) : F � G ∈ FsL(G),Nxy ≤ F � G} (since (G, ·,N) ∈ |SL-NS|, applying the Definition 10(LNGM), and due to Lemma 1, F � G ∈ FsL(G)) = ∨ {H(f←(µ)) : H ∈ FsL(G),Nxy ≤ H} = f←(µ)(xy). Finally, since f←(µ) ≥ f←(µ), we get f←(µ)(x−1) ≥ f←(µ)(x), for any x ∈ G. In fact, for any µ ∈ L(H), f←(µ) ∈ L(G) by Theorem 1.2.11[24]. So, we have: f←(µ)(x) = ∨ {F(f←(µ)) : F ∈ FsL(G),F ≥ Nx} ≤ ∨ {F−1(f←(µ)) : F−1 ∈ FsL(G),F−1 ≥ (Nx)−1} ≤ ∨ {F−1(f←(µ)) : F−1 ∈ FsL(G),F−1 ≥ Nx−1} (by Definition 10(LNGI)) = ∨ {G(f←(µ)) : G ∈ FsL(G),G ≥ Nx−1} = f←(µ)(x−1). Lemma 9. [2] Let (G, ·,∆) ∈ |SL-TOPGRP|, µ ∈ ∆ and ν ∈ LG. Then µ · ν ∈ ∆. Proof. Let x ∈ G, µ ∈ ∆ and ν ∈ LG. Then µ · ν(x) = ∨ st=x µ(x) ∗ ν(t) =∨ t∈G µ(xt−1) ∗ ν(t) = ∨ t∈GRt(µ)(x) ∗ ν(t). Fix t ∈ G, then ν(t) is constant and T M G Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 14 (3) (2021), 949-968 965 ν(t) ∈ L. Since and Rt : G −→ G is a homeomorphism, and µ ∈ ∆, ∨ t∈GRt(µ) ∈ ∆ and since ∆ is stratified, and (L, ∗) is commutative semigroup, we have ∨ t∈GRt(µ) ∗ ν(t) = ν(t) ∗ ∨ t∈G Lt(µ) ∈ ∆, i.e., µ · ν ∈ ∆. Proposition 3. [18] Let (X,∆N) be a stratified L-valued topological space with a corre- sponding stratified L-valued neighborhood system N. Then (X,∆N) is Hausdorff-separated if and only if for all x 6= y ∈ X there are ν1, ν2 ∈ ∆N such that ν1 ∗ ν2 = >∅ and ν1(x) ∗ ν2(y) 6= ⊥. Definition 18. [18] Let (X,∆) be a stratified L-valued topological space, N = (Nx)x∈X be the corresponding L-valued neighborhood system, and A be a subset of X. Then closure of A, written as A, is given by A = {x ∈ X : Nx(>X∩Ac) = ⊥} A subset of X is said to be closed with respect to ∆ if A = A. Lemma 10. A stratified L-valued topological group (G, ·,∆N) is Hausdorff-separated if and only if some singleton {a} ⊆ G is closed. In particular {e} is a closed subgroup of G. Proof. Let {a} ⊆ G be closed subset ofG. Then since the mapping ϕ : (G×G,∆×∆)→ (G,∆), (g, h) 7−→ g−1ha is continuous, we have ϕ−1({a}) = {(g, g) : g ∈ G} ⊆ G×G, the diagonal which in view of the Corollary 6.2.1.2 [18], is a closed subset of G×G with respect to the product stratified L-topology ∆×∆ implying that (G, ·,∆) is Hausdorff-separated. Conversely, let x 6∈ {a}. Then x 6= a ∈ X yields that there are ν1, ν2 ∈ ∆ such that ν1 ∗ ν2 ≤ >X∩{a}c and Nx(ν1) ∗Na(ν2) 6= ⊥, which implies that x 6∈ {a}. Lemma 11. If (G, ·,∆N) is a Hausdorff-separated stratified L-valued topological group, and A be a closed subgroup of G, then the normalizer of A in G: NG(A) = {g ∈ G : γa(A) = A} is a closed subgroup of G, where γa : G −→ G defined by γa(g) = ag−1a the conjugation map. Proof. If a ∈ A, take ca(g) = gag−1. Then the mapping ca : G −→ G is continuous and hence the inverse image of the closed set A: c−1 a (A) = {g ∈ G : gag−1 ∈ A} is closed. Thus, we have B := ∧ a∈A c −1 a (A) = {g ∈ G : γa(A) ⊆ A} is a closed subset of G. Since the inversion mapping  : G −→ G, g 7−→ g−1 is a homeo- morphism, A−1 is closed, since A is closed, and hence NG(A) = B ∩A−1 is closed. Lemma 12. Let (G, ·,∆) be a stratified L-valued topological group, N be a corresponding stratified L-valued neighborhood system on G and A is a subset of G. Then the centralizer ZG(A) = {g ∈ G : [g, a] = e ∀a ∈ A} is closed with respect to ∆. In particular, the center of G is closed subgroup. REFERENCES 966 Proof. If a ∈ A, then the mapping ϕ : G −→ G, g 7−→ [g, a] = gag−1a−1 is continuous, where the element of the type gag−1a−1 is called commutator of the group G. Now since {e} is closed subset of G, and since the inverse image of closed subsets under continuous mapping are again closed, in view of the Corollary 6.2.1.2 [18], ZG(a) = {g ∈ G : [g, a] = e} is closed, and as the ZG(A) = ∧ a∈A ZG(a) is closed, hence the result follows. 6. Conclusion In this article, as a continuation of our previous work on L-valued topological groups, where the underlying lattice L was an enriched cl-premonoid, we have presented two types of results, one is about the relationship between L-valued topological groups and their corresponding Kent convergence groups and conversely; the other is about L-valued closure of L-valued subgroup of a group. 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