Algebraic structures of fuzzy T -Locality spaces: Part 1

Abstract

The aim of this paper is to introduce the concepts of fuzzy T-locality groups. These constructs mainly will deal and relate with both fuzzy T-locality spaces (1995) and fuzzy TL-uniform spaces (2006). We establish some basic results and characterization theorems of fuzzy T-locality groups. Also, we give the necessary and the sufficient conditions for a group structure to be compatible with a fuzzy T-locality system.

1 Introduction

In [10], Morsi introduced the notions of fuzzy T-locality spaces, for each continuous triangular norm T. In [6], Hashem and Morsi deduced the concepts of fuzzy TL-uniform spaces which compatible with fuzzy T-locality spaces. In [1], Ahsanullah and Al-Thukair deduced the notions of T-neighbourhood groups.

In this manuscript, we introduce the concepts of fuzzy T-locality groups. This structure is closed connected with fuzzy T-locality spaces. We give some important results of fuzzy T-locality groups and we prove that the left and the right translations of fuzzy T-locality group are homeomorphisms. Also, we generalize two important characterization theorems, which give the necessary and the sufficient conditions for a group structure and a fuzzy T-locality system to be compatible.

We proceed as follows:

In Section 1, we present some basic definitions of fuzzy sets, I-topological spaces, fuzzy T-locality spaces and fuzzy TL-uniform spaces, which are needed throughout this paper.

In Section 2, we deduce some important properties of fuzzy T-locality spaces.

In Section 3, we introduce the concepts of fuzzy T-locality groups and we prove some of their properties. We give the notions of left and the right translations of fuzzy T-locality group. Also, we study the relations between fuzzy T-locality groups and fuzzy TL-uniform spaces.

The motivation of manuscript: The concepts of I-topological groups play an important role in the algebraic I-topology. Here, we introduce the concepts of fuzzy T-locality groups.

1.1 Preliminaries

A triangular norm T (cf. [12]) is a binary operation on the unit interval I = [0, 1] that is associative, symmetric, isotone in each argument and has neutral element 1. The basic three (continuous) triangular norms are their simplest, namely Min (also denoted by ∧), product (π) and the Lukasiewicz conjunction (T_m), with that for all α, β ∈ I: $α π β = α β and α T_{m} β = (α + β) \underline{\land} 1,$

where the binary operation $\underline{\land}$ , above, is the truncated subtraction, defined on non-negative real numbers by: $r \underline{\land} s = max {r - s, 0}, r, s \geq 0$ .

A continuous triangular norm T is uniformly continuous, that is for all ɛ > 0 there is θ = θ_T,ɛ > 0 such that for every (α, β) ∈ I × I, we have $\begin{matrix} (α T β) - ɛ \leq (α - θ) T (β - θ) \\ \leq α T β \leq (α + θ) T (β + θ) \\ \leq (α T β) + ɛ . \end{matrix}$ (1)

Obviously, for every real numbers r, s ≥ 0, ɛ > 0 and the above θ = θ_T,ɛ, we have $(r T s) \underline{\land} ɛ \leq (r \underline{\land} θ) T (s \underline{\land} θ) .$ (2)

For a continuous triangular norm T the following binary operation on I, J (α, γ) = sup {θ ∈ I : α T θ ≤ γ} , α, γ ∈ I, is called the residual implication of T [9]. For this implication, we shall use the following properties [11], ∀ α, β, θ, λ ∈ I: $J (α, α) = 1 .$ (3) $J (α T β, θ T γ) \geq J (α, θ) T J (β, γ) .$ (4)

A fuzzy set λ in a universe set X, introduced by Zadeh in [13], is a function λ : X → I. The collection of all fuzzy sets of X is denoted by I^X. The height of a fuzzy set λ is the following real number: $hgt λ = sup {λ (x) : x \in X} .$

Ordinary subsets of a universe set X will frequently occur in what follows: We shall often need to consider a subset H ⊆ X as a fuzzy subset of X, said to be a crisp fuzzy subset of X, which we shall denote by the symbol 1_H. We do this by identifying 1_H with its characteristic function, the collection of all crisp fuzzy subsets of X is denoted by 2^X. We also denote the constant fuzzy set of X with value α ∈ I by $\underline{\underline{α}}$ . In particular, for a universe X, we denote by $\underline{\underline{1}}$ (resp. $\underline{\underline{0}}$ ) to the crisp fuzzy subset 1_X (resp. 1_φ).

For a given two fuzzy sets μ, λ ∈ I^X, we denote by μ T λ the following fuzzy set of X : (μ T λ) (x) = μ (x) T λ (x) , x ∈ X .

The degree of containment of μ in λ according to J is the real number in I [3], defined by: $J < μ, λ > = inf_{x \in X} J (μ (x), λ (x)) .$

We follow Lowen’s definition of a fuzzy interior operator on a set X [7]. We may define an I-topology in the usual way, namely assuming a fuzzy set μ to be open if and only if μ° = μ.

We denote this I-topology by τ. The pair (X, ^∘) is called an I-topological space.

A function f : (X, ^∘) = (X, τ) → (Y, ^o∖) = (Y, τ^∖), between two I-topological spaces, is said to be continuous [7]; if f^← (μ) ∈ τ for all μ ∈ τ^∖, where $(f^{\leftarrow} (μ)) (x) = μ (f (x)), \forall x \in X .$

Equivalently, if $f^{\leftarrow} (ν^{o ∖}) \leq [f^{\leftarrow} (ν)]^{o}, for all ν \in I^{Y} .$

It is said to be open if f (λ) ∈ τ^∖ for all λ ∈ τ, where (f (λ)) (y) = sup {λ (x) : x ∈ f^← (y)} , y ∈ Y.

Definition 1.1. [10] The T-saturation operator is the operator ^∼T which sends an I-filterbase $B$ in X [8], to the following subset of I^X $\begin{matrix} B^{\sim} T & = & {μ \in I^{X} : \underset{γ \in I_{1}}{\lor} (\underline{\underline{γ}} T μ_{γ}) \leq μ, \\ where μ_{γ} \in B, \forall γ \in I_{1}}, \end{matrix}$

said to be the T-saturation of $B$ . An I-filterbase $B$ is called T-saturated when $B^{\sim T} = B$ .

The fuzzy T-locality spaces (T-locality spaces, for short) were introduced by N. N. Morsi, for more definitions and properties, we can refer to [10].

Definition 1.2 [10]. A T-locality space is an I-topological space (X, ^∘) whose fuzzy interior operator is induced by some indexed family $B = (B (x))_{x \in X}$ , of I-filterbases in X, in the following manner: $\begin{matrix} μ^{\circ} (x) = sup J < ν, μ >, μ \in I^{x}, x \in X . \\ ν \in B (x) \end{matrix}$ (5)

The family $B$ is said to be a T-locality basis for (X, ^∘), and $B^{\sim T}$ is called a T-locality system of (X, ^∘). The I-topology of (X, ^∘) will be denoted by $τ (B)$ . Also, a T-locality base $B$ and a T-locality system $B^{\sim T}$ induce the same T-locality space, that is $τ (B) = τ (B^{\sim T})$ .

Theorem 1.1. [10] A family ofI-filterbases inX, $B = (B (x))_{x \in X}$ , will be aT-locality base in X if and only if it satisfies the following two conditions, for allx ∈ X : $(TLB 1) v (x) = 1 for all ν \in B (x) .$

(TLB2) Every $ν \in B (x)$ has a T-kernel. This consists of two families $(ν_{γ} \in B (x))_{γ \in I_{1}}$ and $(ν_{y γ θ} \in B (y))_{(y, γ, θ) \in X \times} I_{1} \times I_{0}$ such that for all (y, γ, θ) ∈ X × I₁ × I₀, we have $[(γ T ν_{γ} (y)) \underline{\land} θ] T ν_{y γ θ}$ ≤ν.

Theorem 1.2. [10] Let (X, ^∘) and (Y, ^o∖) beT-locality spaces withT-locality bases $B$ andɛ, respectively, and x ∈ X. Then the functionf : (X, ^∘) → (Y, ^o∖) will be continuous at the point x, if and only if for all if and only if for allρ ∈ ɛ (f (x)) and allγ ∈ I₁there is $ρ_{γ} \in B (x)$ such that $\underline{\underline{γ}} T ρ_{γ} \leq f^{\leftarrow} (ρ)$ .

It follows that f will be continuous if it is continuous at all points of its domain.

In [4], Höhle defines for every ψ, φ ∈ I^x×x and λ ∈ I^X:

The T-section of ψ over λ by

$(ψ < λ >_{T}) (x) = sup_{z \in X} [λ (z) T ψ (z, x)], x \in X$

The T-composition of ψ, φ by

$(ψ o_{T} φ) (x, y) = sup_{z \in X} [φ (x, z) T ψ (z, y)], x, y \in X$ .

The symmetric of ψ by _sψ (x, y) = ψ (y, x) , x, y ∈ X.

The fuzzy TL-uniform spaces (TL-uniform spaces, for short) were introduced by K. A. Hashem and N. N. Morsi, for more definitions and properties, we can refer to [6].

Definition 1.4. [6] (i) A TL-uniform base on a set X is a subset υ ⊂ I^x×x which fulfills the following properties:

(TLUB1) υ is an I-filterbase;

(TLUB2) For all φ ∈ υ and x ∈ X, we have ψ (x, x) =1;

(TLUB3) For all φ ∈ υ and γ ∈ I₁, there is φ_γ ∈ υ with $\underline{\underline{γ}} T (φ_{γ} o_{T} φ_{γ}) \leq φ$ ;

(TLUB4) For all φ ∈ υ and γ ∈ I₁, there is φ_γ ∈ υ with $\underline{\underline{γ}} T φ_{γ} \leq_{s} φ$ .

(ii) A TL-uniformity on X is a T-saturated TL-uniform base on X.

(iii) If μ is a TL-uniformity on X, then we shall say that υ is a basis for μ when υ is an I-filterbase and υ^∼T = μ.

It follows that for a TL-uniformity μ on a set X and all φ ∈ μ, we find that _sφ ∈ μ.

The pair (X, μ) consisting of a set X and a TL-uniformity μ on X is called TL-uniform space.

Definition 1.5. [6] Let (X, μ) and (Y, W) be TL-uniform spaces, with bases υ and υ^∖, respectively, and f : X → Y be a function. We say that f is uniformly continuous if for every φ ∈ υ^∖ and γ ∈ I₁, there is ψ ∈ υ such that $\underline{\underline{γ}} T ψ \leq (f \times f)^{\leftarrow} (φ)$ .

Proposition 1.1. [6] Ifμis aTL-uniformity on a set X, then the indexed family (μ (x))_x∈Xgiven byμ (x) = {ψ < 1_x >_T : ψ ∈ μ} is aT-locality system onX.

2 Results on T-locality spaces

In this section, we deduce some important properties of T-locality spaces.

Proposition 2.1.Let $(X, O_{1}, τ (B_{1}))$ and (Y, ^O₂, $τ (B_{2}))$ be twoT-locality spaces with basis $B_{1} = (B_{1} (x))_{x \in X}$ and $B_{2} = (B_{2} (y))_{y \in Y}$ in X andY, respectively. Then their T-product (X × Y, is a T-locality space with a base $B = B_{1} \otimes_{T} B_{2}$ , defined by $B (x, y) = {ν_{1} \otimes_{T} ν_{2} : ν_{1} \in B_{1} (x), ν_{2} \in B_{2} (y)}$ , where (ν₁ ⊗_T ν₂) (x, y) = ν₁ (x) T ν₂ (y), for every (x, y) ∈ X × Y. Moreover, (λ ⊗_T μ) ^o ≥ λ^O1 ⊗_T μ^O2for allλ ∈ I^X, μ ∈ I^Y.

Proof. First, we show that for every (x, y) ∈ X × Y, $B (x, y)$ is an I-filterbase:

Obviously, $B \neq \emptyset$ and $\underline{\underline{0}} \notin B$ .

Let λ₁, $λ_{2} \in B (x, y)$ . Then there are $ν_{1}, ν_{2} \in B_{1} (x)$ and $μ_{1}, μ_{2} \in B_{2} (y)$ such that λ₁ = ν₁ ⊗_T μ₁ and λ₂ = ν₂ ⊗_T μ₂. So, for every (x, y) ∈ X × Y, we have $\begin{matrix} (λ_{1} \land λ_{2}) (x, y) = λ_{1} (x, y) \land λ_{2} (x, y) \\ = (ν_{1} \otimes_{T} μ_{1}) (x, y) \land (ν_{2} \otimes_{T} μ_{2}) (x, y) \\ = [ν_{1} (x) T μ_{1} (y)] \land [ν_{2} (x) T μ_{2} (y)] \\ \geq [ν_{1} (x) \land ν_{2} (x)] T [μ_{1} (y) \land μ_{2} (y)], clear \\ = (ν_{1} \land ν_{2}) (x) T (μ_{1} \land μ_{2}) (y) \\ \geq ν (x) T μ (y), by hypothesis, where \\ ν \in B_{1} (x) and μ \in B_{2} (y) \\ = (ν \otimes_{T} μ) (x, y) \\ = λ (x, y), where λ = ν \otimes_{T} μ \in B (x, y) \end{matrix}$

Hence, the intersection of any two members of $B (x, y)$ contain a member of $B (x, y)$ , which proving that $B (x, y)$ is an I-filterbase in X × Y.

Now, we fulfill the conditions of Theorem 1.1:

(TLB1) For every $λ \in B (x, x)$ and (x, x) ∈ X × Y,

we have $\begin{matrix} λ (x, x) = (ν \otimes_{T} μ) (x, x), ν \in B_{1} (x), μ \in B_{2} (x) \\ = ν (x) T μ (x) \\ = 1 T 1, by hypothesis \\ = 1 . \end{matrix}$

(TLB2) Let $λ \in B (x, y)$ and (x, y) ∈ X × Y. Then there are $ν \in B_{1} (x)$ and $μ \in B_{2} (x)$ such that λ = ν ⊗_T μ.

Now, since ν has a T-kernel, that is a two families $(ν_{γ} \in B_{1} (x))_{γ \in} I_{1}$ and $(ν_{z γ θ} \in B_{1} (z))_{(z, γ, θ) \in X \times} I_{1} \times I_{0}$ such that for all (z, γ, θ) ∈ X × I₁ × I₀, $[(γ T ν_{γ} (z)) \underline{\land} θ] T ν_{z γ θ} \leq ν$ .

Also, since μ has a T-kernel, that is a two families $(μ_{γ} \in B_{2} (y))_{γ \in} I_{1}$ and $(μ_{s γ θ} \in B_{2} (s))_{(s, γ, θ) \in Y \times} I_{1} \times I_{0}$ such that for all (s, γ θ) ∈ Y × I₁ × I₀, we have $[(γ T μ_{γ} (s)) \underline{\land} θ] T μ_{s γ θ} \leq μ$ .

Hence, for every α ∈ I₁ and ɛ ∈ I₀, we can get, by continuity of T and γ ∈ I₁ such that α = γ T γ and then θ = θ_T,ɛ be as in (1).

By taking $λ_{α} = ν_{γ} \otimes_{T} μ_{γ} \in B (x, y)$ and λ_zsαɛ $= ν_{z γ θ} \otimes_{T} μ_{s γ θ} \in B (x, y)$ , we have $\begin{matrix} [(α T λ_{α} (z, s)) \underline{\land} ɛ] T λ_{zs α ɛ} \\ = & {[γ T γ T (ν_{γ} \otimes_{T} μ_{γ}) (z, s)] \underline{\land} ɛ} T λ_{zs α ɛ} \\ = & {[γ T γ T (ν_{γ} \otimes_{T} μ_{γ}) (z, s)] \underline{\land} ɛ} T λ_{zs α ɛ} \\ = & {[γ T γ T ν_{γ} (z) T μ_{γ} (s)] \underline{\land} ɛ} T \\ {ν_{z γ θ} \otimes_{T} μ_{s γ θ}} \\ \leq & [(γ T ν_{γ} (z)) \underline{\land} θ] T [(γ T μ_{γ} (s)) \\ \underline{\land} θ] T [ν_{z γ θ} \otimes_{T} μ_{s γ θ}],, by (2) \\ = & {[(γ T ν_{γ} (z)) \underline{\land} θ] T ν_{z γ θ}} \otimes_{T} \\ {[(γ T μ_{γ} (s)) \underline{\land} θ] T μ_{S γ θ}}, clear \\ \leq & ν \otimes_{T} μ = λ . \end{matrix}$

This proves λ has a T-kernel and thus $B$ satisfies (TLB2).

In order to prove the final part, we follows that:

Let λ ∈ I^X, μ ∈ I^Y and (x, y) ∈ X × Y.

Then in view of Definition 1.2, we have

$\begin{matrix} (λ \otimes_{T} μ)^{o} (x, y) = sup sup \\ ν \in B (x) ϑ \in B (y) \\ J < ν \otimes_{T} ϑ, λ \otimes_{T} μ_{γ} > \\ = sup sup inf inf \\ ν \in B (x) ϑ \in B (y) z \in X s \in Y \end{matrix}$

$\begin{matrix} J ((ν \otimes_{T} ϑ) (z, s), (λ \otimes_{T} μ) (z, s)) \\ = sup sup inf inf J (ν (z) T \\ ν \in B (x) ϑ \in B (y) z \in X s \in Y \\ J (ν (z) T ϑ (s), λ (z) T μ (s)) \\ \geq \sup \sup \inf \inf \\ ν \in B (x) ϑ \in B (y) z \in X s \in Y \\ [J (ν (z), λ (z)) T J (ϑ (s), μ (s))] \\ = [\sup \inf J (ν (z), λ (z))] T \\ ν \in B (x) z \in X \\ [\sup \inf J (ϑ (s), μ (s))] \\ ϑ \in B (y) s \in Y \\ = λ^{O_{1}} (x) T μ^{O_{2}} (y) \\ = (λ^{O_{1}} \otimes_{T} μ^{O 2}) (x, y) . \end{matrix}$

That is, (λ ⊗_T μ) ^o ≥ λ^{O
₁} ⊗_T μ^{O
₂}.

This completes the proof. □

Proposition 2.2.Let $(X, τ (B_{1}))$ and $(Y, τ (B_{2}))$ beT-locality spaces. Then the projections ${Pr}_{1} : (X \times Y, τ (B_{1}) \otimes_{T} τ (B_{2})) \to (X, τ (B_{1}));$ ${Pr}_{2} 1 : (X \times Y, τ (B_{1}) \otimes_{T} τ (B_{2})) \to (Y, τ (B_{2}));$

are continuous.

Proof. Let $ρ \in B_{1} (x)$ and γ ∈ I₁. Then $\begin{matrix} ({Pr}_{\overset{\leftarrow}{1}} (ρ)) (x, y) = ρ ({Pr}_{1} (x, y)) \\ = ρ (x) \\ \geq ρ (x) T ν (y), for some ν \in B_{2} (y) \\ = (ρ \otimes_{T} ν) (x, y) \\ \geq [\underline{\underline{γ}} T (ρ \otimes_{T} ν)] (x, y) . \\ = (\underline{\underline{γ}} T ρ_{γ}) (x, y), \\ ρ_{γ} = ρ \otimes_{T} ν \in B_{1} (x) \times B_{2} (y) . \end{matrix}$

That is, $\underline{\underline{γ}} T ρ_{γ} \leq {Pr}_{\overset{\leftarrow}{1}} (ρ)$ .

Which proves, by Theorem 1.2, the continuity of Pr₁. Similarly we prove the continuity of Pr₂. □

3 Fuzzy T-locality groups

In this section, we introduce the concepts of fuzzy T-locality groups and we show that the left and the right translations of fuzzy T-locality group are homeomorphisms. Moreover, we study some relations between fuzzy T-locality groups and fuzzy TL-uniform spaces. We also show that every fuzzy T-locality group is TL-uniformizable.

In the following, we consider (G, *) as a group with e as the identity element and for every λ : G → I, we define _sλ : G → I, as _sλ (x) = λ (x^-1), for each x ∈ G, where x^-1 is the inverse element of x.

Definition 3.1. Let (G, *) be a group and $(G, τ (B))$ a T-locality space with base $B$ on X. Then the triple $(G, *, τ (B))$ is called a (fuzzy) T-locality group if the following mappings $Γ : (G \times G, τ (B) \otimes_{T} τ (B)) \to (G, τ (B))$ defined by Γ (x, y) = x y, for all x, y ∈ G; $γ : (G, τ (B)) \to (G, τ (B))$ defined by γ (x) = x^-1, for all x ∈ G, are continuous.

Theorem 3.1.If (G, *) is a group, then the triple $(G, *, τ (B))$ is aT-locality group if and only if the mapping $Ω : (G \times G, τ (B) \otimes_{T} τ (B)) \to (G, τ (B))$ de fined byΩ (x, y) = x y^-1, for all x, y ∈ G; is continuous.

Proof. Let $(G, *, τ (B))$ be a T-locality group and $h : (G \times G, τ (B) \otimes_{T} τ (B)) \to (G \times G, τ (B)$ $\otimes_{T} τ (B))$ a mapping defined by h (x, y) = (x, y) ^-1. Then h is the product of the identity mapping I_G and the continuous mapping γ, therefore h obviously is continuous. Hence, Ω = Γ o h is the composition of continuous mappings Γ and h. Consequently, Ω is continuous.

On the other hand, let Ω be continuous mapping and $i : (G, τ (B)) \to (G \times G, τ (B) \otimes_{T} τ (B))$ the canonical injection defined by i (x) = (e, x), where e is the identity element of G. Then γ = Ω o i is the composition of continuous map pings and therefore is continuous.

Since Γ = Ω o h and $h = I_{G} \times γ$ is the product of continuous mappings $I_{G}$ and γ. Thus h is continuous and therefore, Γ also is continuous.

Hence $(G, *, τ (B))$ is a T-locality group. □

For a group (G, *) and _sλ, ν : G → I, we define (λ Θ_T ν) : G → I, by: $\begin{matrix} (λ Θ_{T} ν) (x) = sup_{yz = x \in G} [λ (y) T ν (z)], \\ for every x \in G . \end{matrix}$

Lemma 3.1.If (G, _*) is a group andλ : G → I, then for all x, y ∈ G, we have $\begin{matrix} (1_{x} Θ_{T} λ) (y) = λ (x^{- 1} y) and \\ (λ Θ_{T} 1_{x}) (y) = λ (y x^{- 1}) . \end{matrix}$

Proof. Let λ : G → I and x, y ∈ G. Then $\begin{matrix} (1_{x} Θ_{T} λ) (y) = sup_{zs = y \in G} [1_{x} (z) T λ (s)] \\ = sup_{xs = y \in G} λ (s) \\ = sup_{s = x^{- 1} y \in G} λ (s) \\ = λ (x^{- 1} y) . \end{matrix}$

Analogously, we show (λ Θ_T 1_x) (y) = λ (y x^-1).

This completes the proof. □

For each a group (G, _*) and α ∈ G, the left and the right translations are the following homomorphisms L_α : (G, _*) → (G, _*) , defined by L_α (x) = α x and R_α : (G, _*) → (G, _*) , defined by R_α (x) = x α, for every x ∈ G, respectively.

The left and the right translations in T-locality groups fulfill the following properties.

Proposition 3.1.Let $(G, *, τ (B))$ be aT-locality group. Then for eachα ∈ G, we have

L_α and R_α are homeomorphisms;

(1_α Θ_T λ) = L_α (λ) and (λ Θ_T 1_α) = R_α (λ), for every λ∈ I^G ;

$ν \in (B (e))^{\sim T}$ if and only if $L_{α} (ν) \in (B (α))^{\sim T}$ if and only if $R_{α} (ν) \in (B (α))^{\sim T};$

$ν \in (B (e))^{\sim T}$ if and only if $L_{α^{- 1}} (ν) \in (B (e))^{\sim T}$ if and only if $R_{α^{- 1}} (ν) \in (B (e))^{\sim T};$

If $B$ is T-saturated, then $λ \in B (e)$ if and only if $(1_{α} Θ_{T} λ) \in B (α);$

Proof. (i) The left translation L_α is the composition of the mapping Γdefined above and the injection map $i : (G, τ (B)) \to (G \times G, τ (B) \otimes_{T} τ (B))$ defined by i (x) = (e, x), that is L_α = Γ o i .

Hence, L_α is continuous and bijective. Since (L_α) ^← = L_{α
^-1}, then (L_α) ^← is also continuous.

Consequently, L_α is a homeomorphism.

Similarly, we prove R_α is a homeomorphism.

(ii) Let λ ∈ I^G and α ∈ G. Then for every y ∈ G, we have $\begin{matrix} (L_{α} (λ)) (y) = sup {λ (z) : z \in (L_{α})^{\leftarrow} (y)} \\ = sup {λ (z) : z \in L_{α^{- 1}} (y)} \\ = sup {λ (z) : z \in α^{- 1} (y)} \\ = λ (α^{- 1} y) \\ = (1_{α} Θ_{T} λ) (y), by Lemma 3 . 1 . \end{matrix}$

That is, (1_α Θ_T λ) = L_α (λ).

Similarly, we prove (λ Θ_T 1_α) = R_α (λ).

(iii) Let $ν \in (B (e))^{\sim T}$ . Then for every γ ∈ I₁, there is $ν_{γ} \in B (e) = B (α^{- 1} α) = B (L_{α^{- 1}} (α))$ such that $\underline{\underline{γ}} T ν_{γ} \leq ν$ .

Since L_{α
^-1} is continuous, then in view of Theorem 1.2, we get for every θ ∈ I₁, there is $ν_{γ θ} \in B (α)$ such that $\underline{\underline{θ}} T ν_{γ θ} \leq (L_{α^{- 1}})^{\leftarrow} (ν_{γ}) = L_{α} (ν_{γ})$ . Thus

$\begin{matrix} \underline{\underline{γ}} T \underline{\underline{θ}} T ν_{γ θ} \leq \underline{\underline{γ}} T L_{α} (ν_{γ}) \\ = L_{α} (\underline{\underline{γ}} T ν_{γ}) . \\ \leq L_{α} (ν) \end{matrix}$

Putting β = (γ T θ) ∈ I₁ and $ν_{β} = ν_{γ θ} \in B (α)$ , we have $\underline{\underline{β}} T ν_{β} \leq L_{α} (ν)$ , which implies $L_{α} (ν) \in (B (α))^{\sim T}$ .

Conversely, let $L_{α} (ν) \in (B (α))^{\sim T}$ . Then for every γ ∈ I₁ there is $ν_{γ} \in B (α) = B (α e) = B (L_{α} (e))$ such that $\underline{\underline{γ}} T ν_{γ} \leq L_{α} (ν)$ .

Since L_α is continuous, then by Theorem 1.2 again, we get for every θ ∈ I₁, there is $ν_{γ θ} \in B (e)$ such that $\underline{\underline{θ}} T ν_{γ θ} \leq (L_{α})^{\leftarrow} (ν_{γ})$ .

Putting β = (γ T θ) ∈ I₁ and $ν_{β} = ν_{γ θ} \in B (e)$ , we have

$\begin{matrix} \underline{\underline{β}} T ν_{β} = \underline{\underline{γ}} T \underline{\underline{θ}} T ν_{γ θ} \leq \underline{\underline{γ}} T (L_{α})^{\leftarrow} (ν_{γ}) \\ = (L_{α})^{\leftarrow} (\underline{\underline{γ}} T ν_{γ}) \\ \leq (L_{α})^{\leftarrow} (L_{α} (ν)) \\ = v for L_{α} is injection . \end{matrix}$

This implies $ν \in (B (e))^{\sim T}$ .

Analogously, we show $ν \in (B (e))^{\sim T}$ if and only if $R_{α} (ν) \in (B (α))^{\sim T}$ .

(iv) Follows immediately from (iii).

(v) Let $B$ be a T-saturated I-filterbase. Then $λ \in B (e)$ $\begin{matrix} iff λ \in B (α^{- 1} α) = B (L_{α^{- 1}} (α)), α \in G \\ iff \exists ν \in B (L_{α^{- 1}} (α)) such that ν \leq λ \\ iff \exists ν \in B (L_{α^{- 1}} (α)), for which \\ (L_{α^{- 1}})^{\leftarrow} (ν) \in (B (α))^{\sim T} and ν \leq λ, \\ by Theorem 1.2 \\ iff \exists ν \in B (L_{α^{- 1}} (α)), for which \\ L_{α} (ν) \in (B (α) and ν \leq λ, by hypohtesis \\ iff \exists ν \in B (L_{α^{- 1}} (α)), for which \\ L_{α} (ν) \in (B (α) and L_{α} (ν) \leq L_{α} (λ) \\ iff L_{α} (λ) \in B (α) \\ iff (1_{α} Θ_{T} λ) \in B (α), by (ii) . \end{matrix}$

This completes the proof. □

Next, we give the characterization theorem of T-locality groups, which shows the necessary and the sufficient conditions of a group structure compatible to a T-locality system.

Theorem 3.2.If (G, _*) is a group, then $(G, *, τ (B))$ is aT-locality group if and only if the following are hold:

(i) for all $ν \in B (e)$ and γ ∈ I₁, there is $ν_{γ} \in B (e)$ , such that $\underline{\underline{γ}} T (ν_{γ} Θ_{T} ν_{γ}) \leq ν$ ;

(ii) for all $ν \in B (e)$ and γ ∈ I₁, there is $ν_{γ} \in B (e)$ , such that $\underline{\underline{γ}} T ν_{γ} \leq s^{ν}$ .

Proof. Let $(G, *, τ (B))$ be a T-locality group.

Then the mapping $Γ : (G \times G, τ (B) \otimes_{T} τ (B)) \to (G, τ (B))$ is continuous at all (x, y) ∈ G × G.

In view of Theorem 1.2, we have for every $ν \in B (e) = B (e e) = B (Γ (e, e))$ and γ ∈ I₁, there is $(ν_{γ} \otimes_{T} ν_{γ}) \in B (e, e)$ such that $\underline{\underline{γ}} T Γ (ν_{γ} \otimes_{T} ν_{γ}) \leq ν$ .

Now, for every x ∈ G, we get $\begin{matrix} ν (x) & \geq & [\underline{\underline{γ}} T Γ (ν_{γ} \otimes_{T} ν_{γ})] (x) \\ = & γ T (Γ (ν_{γ} \otimes_{T} ν_{γ})) (x) \\ = & γ T sup {ν_{γ} (y) T ν_{γ} (z) : (y, z) \in Γ^{\leftarrow} (x)} \\ = & γ T sup {ν_{γ} (y) T ν_{γ} (z) : Γ (y, z) = x} \\ = & γ T sup {ν_{γ} (y) T ν_{γ} (z) : yz = x} \\ = & γ T (ν_{γ} Θ_{T} ν_{γ}) (x) . \end{matrix}$

That is $\underline{\underline{γ}} T (ν_{γ} Θ_{T} ν_{γ}) \leq ν$ , which holds (i).

(ii) This follows in the same way as in (i),

by using the continuity of the mapping $γ : (G, τ (B)) \to (G, τ (B)) .$

Conversely, let the stated conditions be hold. If $ν \in B (Γ (e, e)) = B (e e) = B (e)$ and γ ∈ I₁, then from (i) we can get $ν_{γ} \in B (e)$ satisfies $\underline{\underline{γ}} T (ν_{γ} Θ_{T} ν_{γ}) \leq ν$ .

But as above, we can reach to $\underline{\underline{γ}} T Γ (ν_{γ} \otimes_{T} ν_{γ}) \leq ν$ , which meaning by Theorem 1.2, that the mapping $Γ : (G \times G, τ (B) \otimes_{T} τ (B)) \to (G, τ (B))$ is continuous at (e, e) ∈ G × G, and since the translation L_α is continuous at every element α ∈ G.

Then the continuity of the mapping Ω follows from the following composition: $Ω = L_{α θ^{- 1}} o Γ o (L_{α^{- 1}} \times L_{θ^{- 1}}) : G \times G \to G,$

where (α, θ) → (e, e) → e → α θ^-1, for every (α, θ) ∈ G × G

By Theorem 3.1, we have $(G, *, τ (B))$ is a T-locality group. This completes the proof. □

Proposition 3.3.Let (G, _*) be a group and for allλ ∈ I^G, we defineλ_L, λ_R : G × G → I, byλ_L (x, y) = λ (x^-1y) , λ_R (x, y) = λ (yx^–1) , x, y ∈ G. Then for everyν ∈ I^G, the following hold:

(i) λ_L < ν >_T = ν Θ_Tλ and λ_R < ν >_T = λ Θ_T ν;

(ii) (λ T ν)_L = λ_LTν_L and (λ T ν)_R = λ_R T ν_R;

(iii) (_sλ)_L =_s (λ_L) and (_sλ)_R =_s (λ_R);

(iv) (λ Θ_T ν)_L = ν_L o_T λ_L and (λ Θ_T ν)_R = ν_R o_T λ_R.

Proof. (i) For every ν ∈ I^G and y ∈ G, we have $\begin{matrix} (λ_{L} < ν >_{T}) (y) & = & sup_{z \in G} [ν (z) T λ_{L} (z, y)] \\ = & sup_{z \in G} [ν (z) T λ (z^{- 1} y)] \\ = & sup_{z, s = z^{- 1} y \in G} [ν (z) T λ (s)] \\ = & sup_{zs = y \in G} [ν (z) T λ (s)] \\ = & (ν Θ_{T} λ) (y), and \\ (λ_{R} < ν >_{T}) (y) & = & sup_{z \in G} [ν (z) T λ_{R} (z, y)] \\ = & sup_{z \in G} [ν (z) T λ ({yz}^{- 1})] \\ = & sup_{z, s = {yz}^{- 1} \in G} [ν (z) T λ (s)] \\ = & sup_{z, s = {yz}^{- 1} \in G} [λ (s) T ν (z)] \\ = & sup_{sz = y \in G} [λ (s) T ν (z)] \\ = & (λ Θ_{T} ν) (y) . \end{matrix}$

This proves the required equalities.

(ii) Obviously holds.

(iii) Let λ ∈ I^G and x, y ∈ G. Then $\begin{matrix} (_{s} λ)_{L} (x, y) & =_{s} λ (x^{- 1} y) = λ ((x^{- 1} y)^{- 1}) \\ = λ (y^{- 1} x) = λ_{L} (y, x) \\ =_{s} (λ_{L}) (x, y) . \end{matrix}$

Hence (_sλ)_L =_s (λ_L).

Similarly, we prove (_sλ)_R =_s (λ_R).

(iv) For every λ, ν ∈ I^G and x, y ∈ G, we have (λ Θ_T ν)_L (x, y) = (λ Θ_T ν) (x^-1 y) $\begin{matrix} = & sup_{rs = x^{- 1} y \in G} [λ (r) T ν (s)] \\ = & sup_{(x^{- 1} z) (z^{- 1} y) = x^{- 1} y} \\ [λ (x^{- 1} z) T ν (z^{- 1} y)] \\ = & sup_{z \in G} [λ_{L} (x, z) T ν_{L} (z, y)] \\ = & (ν_{L} o_{T} λ_{L}) (x, y), and \\ (λ Θ_{T} ν)_{R} (x, y) & = & (λ Θ_{T} ν) (y x^{- 1}) \\ = & sup_{rs = {yx}^{- 1} \in G} [λ (r) T ν (s)] \\ = & sup_{({zx}^{- 1}) ({yz}^{- 1}) = {yx}^{- 1}} \\ [λ (z x^{- 1}) T ν (y z^{- 1})] \\ = & sup_{z \in G} [λ_{R} (x, z) T ν_{& R} (z, y)] \\ = & (ν_{R} o_{T} λ_{R}) (x, y) . \end{matrix}$

Rendering (iv) and completes the proof. □

In the following we give the relations between T-locality groups and TL-uniform spaces.

Theorem 3.3.EveryT-locality group induces twoTL-uniform spaces.

Proof. If $(G, *, τ (B))$ is a T-locality group, then $(G, τ (B))$ is a T-locality space with T-locality basis $B = (B (x))_{x \in G}$ .

Now, we consider the following collection: $υ = {λ_{L} \in I^{G \times G} : λ \in B (e)}$ , and we claim that υ is a TL-uniform base.

(TLUB1) Obviously υ is an I-filterbase.

(TLUB2) If φ ∈ υ, then there is a $λ \in B (e)$ such that φ = λ_L, and for all x ∈ G, we get, ψ (x, x) = λ_L (x, x) = λ (x^-1 x) = λ (e) =1.

(TLUB3) If φ ∈ υ, then there exists a $λ \in B (e)$ such that φ = λ_L. Thus, by virtue of Theorem 3.2 (i), we have for every γ ∈ I₁, we find $λ_{γ} \in B (e)$ such that $\underline{\underline{γ}} T (λ_{γ} Θ_{T} λ_{γ}) \leq λ$ .

Taking φ_γ = (λ_γ)_L ∈ υ, we have

$\begin{matrix} \underline{\underline{γ}} T (φ_{γ} o_{T} φ_{γ}) = \underline{\underline{γ}} T [(λ_{γ})_{L} o_{T} (λ_{γ})_{L}] \\ = \underline{\underline{γ}} T [(λ_{γ}) Θ_{T} (λ_{γ})_{L}], \end{matrix}$

by Proposition 3.3 (iv) $= [\underline{\underline{γ}} T (λ_{γ} Θ_{T} λ_{γ})]_{L}, clear \leq λ_{L} = φ$ .

(TLUB4) If φ ∈ υ, then there is $λ \in B (e)$ such that φ = λ_L. Consequently, by Theorem 3.2 (ii), we have for every γ ∈ I₁, there exists $λ_{γ} \in B (e)$ such that $\underline{\underline{γ}} T λ_{γ} \leq_{s} λ$ .

Therefore, by Proposition 3.3 (iii), we get $\underline{\underline{γ}} T (λ_{γ})_{L} = (\underline{\underline{γ}} T λ_{γ})_{L} \leq (_{s} λ)_{L} =_{S} (λ_{L})$ , which implies $\underline{\underline{γ}} T φ_{γ} \leq_{s} φ$ .

This shows in accordance with Definition 1.4, that the collection υ is a TL-uniform base, which in turn gives rise to a left TL-uniformity μ = υ^∼T. Analogously, we obtain the right TL-uniformity. This completes the proof. □

Definition 3.3. An I-topological space (X, τ) is called TL-uniformizable if there is a TL-uniformity μ on X such that τ = τ (μ).

Theorem 3.4.EveryT-locality group isTL-uniformizable.

Proof. Let $(G, *, τ (B))$ be a T-locality group. Then $(G, τ (B))$ is a T-locality space with the T-locality system $B^{\sim T} = ((B (x))^{\sim T})_{x \in G}$ .

Now, suppose that (μ (x))_x∈G is a T-locality system associated with the left TL-uniformity μ, then we get μ (x) = {ψ < 1_x >_T : ψ ∈ μ}, by Proposition 1.1 $= {λ_{L} < 1_{x} >_{T} : λ \in (B (e))^{\sim T}}$ , by Theorem 3.3 $= {1_{x} Θ_{T} λ : λ \in (B (e))^{\sim T}}$ , clear $= (B (x)^{\sim T})$ , by Proposition 3.1 (iv).

Therefore, in view of Definition 1.2, we have $τ (B) = τ (B^{\sim T}) = τ (μ)$ .

Which proves $(G, *, τ (B))$ is TL-uniformizable. □

Next, we characterize the uniformly continuous functions between TL-uniform spaces.

Proposition 3.4.Let $(G, *, τ (B))$ and (E, _#, τ (ɛ)) beT-locality groups. Ifυ^∼TandW^∼Tare the associated leftTL-uniformities on G and E, respectively, thenf : (G, υ^∼T) → (E, W^∼T) is uniformly continuous if and only if for allρ ∈ ɛ (e^∖) and γ ∈ I₁, there is $λ \in B (e)$ such that $\underline{\underline{γ}} T (1_{x} Θ_{T} λ) \leq f^{\leftarrow} (1_{f (x)} Θ_{T} ρ)$ , for each x ∈ G, wheree and e^∖ are the identity elements of G and E, respectively.

Proof. In view of Definition 1.5, we have

f : (G, υ^∼T) → (E, W^∼T) is uniformly continuous $\begin{matrix} iff \forall φ \in W, γ \in I_{1} \exists ψ \in υ such that \\ \underline{\underline{γ}} T ψ \leq (f \times f)^{\leftarrow} (φ) \\ iff \forall ρ \in ɛ (e), γ \in I_{1} \exists λ \in B (e) such that \\ \underline{\underline{γ}} T λ_{L} \leq (f \times f)^{\leftarrow} (ρ_{L}) \\ iff \forall ρ \in ɛ (e^{∖}), γ \in I_{1} \exists λ \in B (e) such that \\ (\underline{\underline{γ}} T λ_{L}) (x, y) \leq ρ_{L} (f (x), f (y)), \forall x, y \in G \\ iff \forall ρ \in ɛ (e^{∖}), γ \in I_{1} \exists λ \in B (e) such that \\ γ T λ (x^{- 1} y) \leq ρ ((f (x))^{- 1} f (y)) \\ \forall x, y \in G \\ iff \forall ρ \in ɛ (e^{∖}), γ \in I_{1} \exists λ \in B (e) such that \\ γ T (1_{x} Θ_{T} λ) (y) \leq (1_{f (x)} Θ_{T} ρ) (f (y)), \\ \forall x, y \in G, \\ by Lemma 3.1 \\ iff \forall ρ \in ɛ (e^{∖}), γ \in I_{1} \exists λ \in B (e) such that \\ \underline{\underline{γ}} T (1_{x} Θ_{T} λ) \leq f^{\leftarrow} (1_{f (x)} Θ_{T} ρ), \\ \forall x \in G \end{matrix}$

This completes the proof. □

Proposition 3.5.Let $(G, *, τ (B))$ and $(E, #, τ (B))$ beT-locality groups. Ifυ^∼TandW^∼Tare the associated leftTL-uniformities on G and E, respectively, then the continuous homomorphismf : G → E is uniformly continuous.

Proof. Let f : G → E be a continuous homomorphism, γ ∈ I₁ and ρ ∈ ɛ (e^∖) = ɛ (f (e)).

Then by Theorem 1.2, there is $λ \in B (e)$ such that $\underline{\underline{γ}} T λ \leq f^{\leftarrow} (ρ)$ and hence, we have for every x, y ∈ G, that $\begin{matrix} [\underline{\underline{γ}} T (1_{x} Θ_{T} λ)] (y) & = & γ T (1_{x} Θ_{T} λ) (y) \\ = & γ T λ (x^{- 1} y), by Lemma 3.1 \\ \leq & (f^{\leftarrow} (ρ)) (x^{- 1} y) \\ = & ρ (f (x^{- 1} y)) \\ = & ρ (f (x^{- 1}) f (y)), \\ for f homomorphism \\ = & ρ ((f {(x))}^{- 1} f (y)), clear \\ = & (1_{f (x)} Θ_{T} ρ) (f (y)), \\ by Lemma 3.1 again \\ = & [f^{\leftarrow} (1_{f (x)} Θ_{T} ρ)] (y) . \end{matrix}$

That is $\underline{\underline{γ}} T (1_{x} Θ_{T} λ) \leq f^{\leftarrow} (1_{f (x)} Θ_{T} ρ)$ , which proves that, f is uniformly continuous. □

Now, we show that every group induces a T-locality space.

Theorem 3.5.Let (G, _*) be a group and consider a collectionF ⊂ I^G, which satisfies:

F is an I-filterbase;

For all λ ∈ F, we have λ (e) =1;

For all λ ∈ F, we get _sλ ∈ F;

For all λ ∈ F and γ ∈ I₁, there is λ_γ ∈ F, such that $\underline{\underline{γ}} T (λ_{γ} Θ_{T} λ_{γ}) \leq λ$ .

If we take $B (x) = {(1_{x} Θ_{T} λ) \in I^{G} : λ \in F}$ for every x ∈ G, then $B = (B (x))_{x \in X}$ is a T-locality base on G.

Proof. First, we show that $B = (B (x))_{x \in X}$ is a T-locality base on G, as follows:

$B (x)$ is an I-filterbase, because $\underline{\underline{0}} \notin B (x)$ and for any two members ν₁, ν₂ of $B (x)$ , we have ν₁ ∧ ν₂ = (1_x Θ_T λ₁) ∧ (1_x Θ_T λ₂), for some $\begin{matrix} λ_{1}, λ_{2} \in F = 1_{x} Θ_{T} (λ_{1} \land λ_{2}) \\ \geq (1_{x} Θ_{T} λ), for some λ \in F, since F is an I - \end{matrix}$ filterbase =ν. That is (ν₁ ∧ ν₂) contains a member ν of $B (x)$ .

(TLB1) For every ν ∈ B (x) and x ∈ G, we have ν (x) = (1_x Θ_T λ) (x), for some λ ∈ F= λ (x^-1 x) , by Lemma 3.1 = λ (e) = 1, by (ii).

(TLB2) Let $ν \in B (x)$ . Then for every z ∈ G,we have ν (z) = (1_x Θ_T λ) (z), for some λ ∈ F= λ (x^-1z), by Lemma 3.1 $\geq [\underline{\underline{γ}} T (λ_{γ} Θ_{T} λ_{γ})]$ (x^-1z) , by (iv) $\begin{matrix} = γ T sup_{y \in G} [λ_{r} (x^{- 1} y) T λ_{γ} (y^{- 1} z)] \\ \geq γ T sup_{y \in G} {λ_{r} (x^{- 1} y) T [\underline{\underline{α}} T (λ_{γ α} \\ Θ_{T} λ_{γ α})] (y^{- 1} z)}, by (iv) again \\ = γ T α T sup_{y \in G} {λ_{γ} (x^{- 1} y) T \\ sup_{rs = y^{- 1} z \in G} [λ_{γ α} (r) T λ_{γ α} (s)]} \\ \geq γ T α T λ_{γ} (x^{- 1} y) T λ_{γ α} (y^{- 1} z) T λ_{λ α} (e) \\ = γ T α T λ_{γ} (x^{- 1} y) T λ_{γ α} (y^{- 1} z) T 1, by (ii) \\ = γ T α T (1_{x} Θ_{T} λ_{γ}) (y) T (1_{y} Θ_{T} λ_{γ α}) (z), \end{matrix}$

by Lemma 3.1.

Now, for every θ ∈ I₀, we get (by continuity of T) α = α_θ ∈ I₁, such that $\begin{matrix} [γ T α T (1_{x} Θ_{T} λ_{γ}) (y)] \\ \geq [γ T (1_{x} Θ_{T} λ_{γ}) (y)) \underline{\land} θ] \end{matrix}$ , hence by taking $ν_{x γ} = (1_{x} Θ_{T} λ_{γ}) \in B (x)$ and ν_yγθ = (1_y Θ_T λ_γα) ∈ B (y), we get $[(γ T ν_{x γ} (y)) \underline{\land} θ] T ν_{y γ θ} \leq ν$ .

That is ν has a T-kernel, and therefore $B (B (x))_{x \in X}$ is a T-locality base on G. This completes the proof. □

4 Initial and final T-locality groups

In this section, we show that the category ofT-locality groups and their continuous homomorphisms is a topological category, hence all initial and final T-locality groups exist.

We denote by T-LocGrp the category of T-locality groups together with continuous homomorphisms as morphisms. For any class Λ, let ((H_r, τ _r))_r∈Λ be a family of T-locality groups and (f_r)_r∈Λ a family of homomorphisms of a group G into groups H_r. For any T-locality group (G, τ ), the family (f_r : (G, τ ) ⟶ (H_r, τ _r))_r∈Λ is called an initial lift of (f_r : G ⟶ (H_r, τ _r))_r∈Λ in the category T-LocGrp provided that (G, τ ) is the T-locality group for which the following conditions are fulfilled:

All mappings f_r : (G, τ ) ⟶ (H_r, τ _r) are continuous homomorphisms;

For any T-locality group (H, σ) and any mapping f : (H, σ ) ⟶ (G, τ) is continuous homomorphism if and only if for all r ∈ Λ the mappings f_r o f : (H, σ ) ⟶ (H_r, τ _r) are continuous homomorphisms.

By an initial T-locality group we mean theT-locality group which provides an initial lift in the category T-LocGrp.

To proves all initial lifts and all initial T-locality groups exist in T-LocGrp we have to prove first that in the case f_r : G ⟶ H_r is an injective homomorphism for each r ∈ Λ and τ is the initial T-locality of ( τ _r)_r∈Λ with respect to (f_r)_r∈Λ we have (G, τ ) also is a T-locality group.

Consider Λ is any class:

Proposition 4.1.Let ((H_r, σ _r))_r∈Λbe a family ofT-locality groups and for allr ∈ Λ, letf_r : G ⟶ H_rbe an injective homomorphism of a group G into H_r. Let τ be the initialI-topology of ( σ _r)_r∈Λwith respect to (f_r)_r∈Λ.

Then (G, τ ) also is T-locality group.

Proof. Let Ω_G : (G × G, τ × τ) ⟶ (G, τ ) and Ω_Hr : (H_r × H_r, σ _r × σ _r) ⟶ (H_r, σ _r) be defined as in Theorem 3.1.

Since f_r o Ω_G = Ω_Hr o (f_r × f_r) and Ω_Hr, f_r, are continuous, then f_r o Ω_G is continuous.

From condition (ii) of the initial lift in the category of I-topological spaces, it follows that Ω_G is continuous and therefore (G, τ ) is a T-locality group. □

The following theorem shows that the T-locality group mentioned in Proposition 4.1 fulfills the conditions of an initial lift in the concrete category T-LocGrp.

Theorem 4.1.Let ((H_r, σ _r))_r∈Λbe a family ofT-locality groups and for allr ∈ Λ, letf_r : G ⟶ H_rbe an injective homomorphism of a group G into H_rand let τ be the initialI-topology of ( σ _r)_r∈Λwith respect to (f_r)_r∈Λ.

Then (f_r : (G, τ ) ⟶ (H_r, σ _r))_r∈Λ is an initial lift of (f_r : G ⟶ (H_r, σ _r))_r∈Λ in the category T-LocGrp.

Proof. First, Propositions 4.1 shows that (G, τ ) is a T-locality group.

From condition (i) of an initial lift in the category of I-topological spaces, we get that condition (i) of initial lift in T-LocGrp holds, that is, all f_r : (G, τ ) ⟶ (H_r, σ _r) are continuous homomorphisms.

Second, let (H, σ ) be a T-locality group and f be a mapping from H into G.

Then from condition (ii) of an initial lift in the category of I-topological spaces, we get f : (H, σ ) ⟶ (G, τ ) is continuous if and only if all f_r o f : (H, σ ) ⟶ (H_r, σ_r) are continuous.

Now, if f and for all f_r, are homomorphisms, then so is f_r o f. Conversely, let all f_r o f be homomorphisms. Since all are homomorphisms we have for every x, y ∈ H, $\begin{matrix} f_{r} (f (x y)) \\ = (f_{r} o f) (x y) = [(f_{r} o f) (x)] [(f_{r} o f) (y)] \\ = [f_{r} (f (x))] [f_{r} (f (y))] = f_{r} (f (x) f (y)) . \end{matrix}$

Moreover, since f_r is an injective, we get f (x y) = f (x) f (y), that is, f is a homomorphism.

Hence, f : (H, σ ) ⟶ (G, τ ) is continuous homomorphism if and only if all o f_r of : (H, σ ) ⟶ (H_r, σ _r) are continuous homomorphisms, that is, condition (ii) of an initial lift in T-LocGrp is fulfilled. □

Theorem 4.1 states that all initial lifts exist uniquely in the concrete category T-LocGrp and this means that the category T-LocGrp is a topological category. Hence, all initial T-locality groups exist.

By means of Theorem 4.1, the T-locality groups introduced in Proposition 4.1 coincide with the initial T-locality groups, that is, if ((H_r, σ _r))_r∈Λ is a family of T-locality groups, and for each r ∈ Λ, f_r is an injective homomorphism of a group G into H_r and τ is the initial I-topology of ( σ _r)_r∈Λ with respect to (f_r)_r∈Λ, then (G, τ ) is the initial T-locality group.

T-locality subgroups and T-locality product groups are special initial T-locality groups and hence the above implies the following result.

Corollary 4.1. (i) If (G, τ ) is a T-locality group and S a subgroup of G, then the I-topological subspace (G, τ _S) also is T-locality group, called a T-locality subgroup. (ii) If ((G_r, τ _r))_r∈Λ is a family of T-locality groups and G is the product $\underset{r \in Λ}{Π}$ G_r of the family (G_r)_r∈Λ of groups and $τ = \underset{r \in Λ}{Π} τ_{r}$ is the product of the family ( τ _r)_r∈Λ of I-topologies, then (G, τ ) also is a T-locality group, called a T-locality product group.

Now, since the concrete category T-LocGrp is a topological category, then all final lifts also uniquely exist. This even means that also all final T-locality groups exist.

If ((G_r, τ _r))_r∈Λ is a family of T-locality groups and (f_r)_r∈Λ a family of homomorphisms from G_r into a group H, indexed by any class Λ. For any T-locality group (H, σ ), the family (f_r : (G_r, τ _r) ⟶ (H, σ ))_r∈Λ is called a final lift of (f_r : (G_r, τ _r) ⟶ H)_r∈Λ in the category T-LocGrp provided that (H, σ) is the T-locality group which fulfills the following conditions:

All mappings f_r : (G_r, τ_r) ⟶ (H, σ) are continuous homomorphisms;

For any T-locality group (G, τ ) and any mapping f : (H, σ ) ⟶ (G, τ ) is continuous homomorphism if and only if for all r ∈ Λ the mappings f o f_r : (G_r, τ _r) ⟶ (G, τ ) are continuous homomorphisms.

By a final T-locality group we mean a T-locality group which provides a final lift in the category T-LocGrp.

The following propositions show that if for each r ∈ Λ, f_r : (G_r, τ _r) ⟶ (H, σ ) is a surjective homomorphism and σ is the final T-locality of ( τ _r)_r∈Λ with respect to (f_r)_r∈Λ, then (H, σ ) also is a T-locality group. To prove these results we need the following proposition, which can be proved easily by means of the properties of T-locality group.

Proposition 4.3. If f : (G, τ ) ⟶ (H, f ( τ )) is a surjective homomorphism from a T-locality group(G, τ ) into a group H equipped with the final I-topology f ( τ ) of τ with respect to f, then f is an open function.

Consider the case of Λ being a singleton:

Proposition 4.4. Let (G, τ ) be a T-locality group and let f : (G, τ ) ⟶ H be a homomorphism of a group G onto H. Then the final T-locality space (H, f ( τ )) of (G, τ ) with respect to f also is a T-localitygroup.

Proof. Let Ω_G : (G × G, τ × τ) ⟶ (G, τ ) and Ω_H : (H × H, f ( τ ) × f ( τ )) ⟶ (H, f (τ)) be defined as in Theorem 3.1. H Since f is a surjective homomorphism, then for every μ ∈ I^H and X, Y ∈ H wehave $\begin{matrix} (Ω \overset{\leftarrow}{H} (μ)) (X, Y) \\ = μ (Ω_{H} (X, Y)) \\ = {μ (Ω_{H} (f (r), f (s)) : r, s \in G, (f \times f) (r, s) = (x, y)} \\ = \sup {μ (f (r) {(f (s))}^{- 1}) : (r, s) \in {(f \times f)}^{\leftarrow} (x, y)}, clear \\ = \sup {μ (f (r) f (s^{- 1})) : (r, s) \in {(f \times f)}^{\leftarrow} (x, y)} \\ = \sup {μ (f ({rs}^{- 1})) : (r, s) \in {(f \times f)}^{\leftarrow} (x, y)} \\ = \sup {(f^{\leftarrow} (μ)) ({rs}^{- 1}) : (r, s) \in {(f \times f)}^{\leftarrow} (x, y)} \\ = \sup {(f^{\leftarrow} (μ)) (Ω_{G} (r, s)) : (r, s) \in {(f \times f)}^{\leftarrow} (x, y)} \\ = \sup {(Ω_{G}^{\leftarrow} (f^{\leftarrow} (μ))) (r, s) : (r, s) \in {(f \times f)}^{\leftarrow} (x, y)} \\ = [(f \times f (Ω_{G}^{\leftarrow} (f^{\leftarrow} (μ))))] (x, y) . \end{matrix}$

That is, $Ω_{H}^{\leftarrow} (μ) = (f \times f) (Ω_{G}^{\leftarrow} (f^{\leftarrow} (μ))) .$ If μ ∈ f ( τ ), then f^← (μ) ∈ ( τ ) and from continuity of Ω_G we have $Ω_{G}^{\leftarrow} (f^{\leftarrow} (μ)) \in τ \times τ .$

But from Proposition 4.3, we have f is an open, hence f × f : G × G ⟶ H × H is obviously an open. Consequently, $Ω_{H}^{\leftarrow} (μ) = (f \times f) (Ω_{G}^{\leftarrow} (f^{\leftarrow} (μ))) \in f (τ) \times f (τ) .$

Which proves the continuity of Ω_H and this implies (H, f ( τ )) is a T-locality group. □

For any class Λwe have the following:

Proposition 4.5. Let ((G_r, τ _r))_r∈Λ be a family ofT-locality groups and for all r ∈ Λ, let f_r : G_r ⟶ H be a homomorphism of a group G_r onto a group H. If σ is the final I-topology of ( τ _r)_r∈Λ with respect to (f_r)_r∈Λ, then (H, σ ) also is T-locality group.

Proof. Let μ ∈ σ . Since f_r : (G_r, τ _r) ⟶ (H, σ ) is continuous, then $f_{r}^{\leftarrow} (μ) \in τ_{r}$ for all r ∈ Λ. But from continuity of Ω_Gr : (G_r × G_r, τ _r × τ _r) ⟶ (G_r, τ _r), we get $Ω_{G_{r}}^{\leftarrow} (f_{r}^{\leftarrow} (μ)) \in τ_{r} \times τ_{r} .$

By similar proof of Proposition 4.4, we have $Ω_{H}^{\leftarrow} (μ) = (f \times f) (Ω_{G_{r}}^{\leftarrow} (f_{r}^{\leftarrow} (μ))),$ where Ω_H : (H × H, σ × σ ) ⟶ (H, σ ) . Moreover, all f_r × f_r are open, hence $Ω_{H}^{\leftarrow} (μ) \in σ \times σ$ . This proves Ω_H is continuous and thus (H, σ ) is a T-locality group. □

Next, we are going to show that the T-locality group given in Propositions 4.4 and 4.5 fulfills the conditions of a final lift in the concrete category T-LocGrp.

Theorem 4.2. Let ((G_r, τ _r))_r∈Λ be a family of T-locality groups and for all r ∈ Λ, let f_r : G_r ⟶ H be a surjective homomorphism of a group G_r into H and let σ be the final I-topology of ( τ _r)_r∈Λ with respect to (f_r)_r∈Λ. Then (f_r : (G_r, τ _r) ⟶ (H, σ ))_r∈Λ is the final lift of (f_r : (G_r, τ _r) ⟶ H)_r∈Λ in the category T-LocGrp.

The proof goes similarly, using Propositions 4.4 and 4.5 and the properties of final lift in the category T-LocGrp, as in case of Theorem 4.1. □

From Theorem 4.2, we get that the T-locality groups introduced in Propositions 4.4 and 4.5, can be considered as the final T-locality groups.

T-locality quotient group is special final T-locality group and hence the above implies the following result.

Footnotes

Acknowledgments

The author is grateful to an anonymous referee for his/her generous advice that significantly improved the presentation of this article.

References

Ahsanullah

T.M.G.

and Al-Thukair

, T-neighbourhood groups, Int J of Mathematics and Mathematical Sciences14 (2004), 703–719.

Yu Chunhai and Ma jiliang, L-fuzzy topological groups, Fuzzy Sets and Systems44 (1991), 83–91.

Gottwald

, Set theory for fuzzy sets of higher level, Fuzzy Sets and Systems2 (1979), 125–151.

Höhle

, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems8 (1982), 63–69.

jiliang

and Haj

Yu Chan

, Fuzzy topological groups, Fuzzy Sets and Systems12 (1984), 289–299.

Hashem

K.A.

and Morsi

N.N.

, Fuzzy TL-Uniform spaces, International Journal of Mathematics and Mathematical Sciences2006,1–24. Article ID 25094.

Lowen

, Fuzzy topological spaces and fuzzy compactness, J Math Anal Appl56 (1976), 621–633.

Lowen

, Fuzzy uniform spaces, J Math Anal Appl82 (1981), 370–385.

Morsi

N.N.

, Hyperspace fuzzy binary relations, Fuzzy Sets and Systems67 (1994), 221–237.

10.

Morsi

N.N.

, Fuzzy T-locality spaces, Fuzzy Sets and Systems69 (1995), 193–219.

11.

Morsi

N.N.

, A small set of axioms for residuated logic, Information Sciences75 (2005), 85–96.

12.

Schweizer

and Sklar;

, Probabilistic Metric Spaces, North-Holland, Amsterdam1983.

13.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 338–353.