EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1762-1778
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On the Characterizations of Approach Groups

T.M.G. Ahsanullah1,∗, Fawzi Al-Thukair1

1 Department of Mathematics, College of Science, King Saud University, Riyadh,
Saudi Arabia

Abstract. In this paper, we present several characterization theorems on approach groups, and
ultra approach groups. In so doing, we first give necessary and sufficient conditions for an approach
structure to be compatible with group structure. We show that every ultra approach group is ultra-
uniformizable. Secondly, starting with an approach space, and its natural neighborhood system
on a group, we characterize the resulting neighborhood approach group. Finally, we show that
the category of ultra approach-Cauchy group is a topological category, and more importantly, we
show that the category of ultra approach-Cauchy groups and the category of strongly normal ultra
approach-limit groups are isomorphic.

2020 Mathematics Subject Classifications: 54A20, 54E70, 54E90, 54H11

Key Words and Phrases: Approach space, approach group, ultra approach group, approach
neighborhood system, approach limit space, approach limit group, approach uniformity, ultra
approach-Cauchy group, strongly normal ultra approach-Cauchy group, category

1. Introduction

It is observed in [15], and elsewhere that TOP, the category of topological spaces, is si-
multaneously bireflectively and bicoreflectively embedded in AP, the category of approach
spaces. This shows, however, that it makes not much difference notions like limits, colimits,
initial structure that we may consider either in TOP or in AP. But it does make dif-
ference whether we make initial structures of ∞pq-metric approach spaces in pqMET∞,
the set of all ∞pq-metrics or in AP. This is so, because of the facts that in the first place,
the domain of the ordinary metric space object (X, d : X ×X → [0,∞]) belonging to the
category pqMET∞ and the distance space object

(
X,∆d : X × 2X → [0,∞]

)
known as

metric distance space, being member of the category AP, are essentially different; and, in
the second place, they are also different from categorical viewpoint.
Given the importance of the preceding paragraph, a vast scale of research articles appeared
over the years on studying various aspects of approach spaces, and their equivalence struc-
tures. A tiny part of the work cited in this paper cf. [4, 6–8, 13, 14, 17, 18, 20], whereas

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5279

Email addresses: tmga1@ksu.edu.sa (T.M.G. Ahsanullah), thukair@ksu.edu.sa (Fawzi Al-Thukair)

https://www.ejpam.com 1762 © 2024 EJPAM All rights reserved.



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1763

the vast majority of research carried out in these ever-growing area are not mentioned here
just because the present paper is not directly linked to those works. It is pointed out in
[9, 14, 19, 21], the importance of non-archimedean approach structures or ultra approach
structures. We find it interesting to study the compatibility of the non-Archimedean ap-
proach structures with group structures, particularly, ultra approach-Cauchy structures,
ultra approach limit structures, and so on; although we do not intend to study non-
Archimedean metric group or ultra metric group structures here in this paper.
The idea of approach group along with its uniformization first appeared in [18], and later,
we modified this concept further in [4] in a wider context. Furthermore, we identified
this approach group with some other structures cf. [2, 3]. In this paper, we characterize
approach groups vis-à-vis ultra approach groups, and study some of their related results.
Thus, we concentrate on three main issues in relation with approach groups, such as, (a)
ultra approach groups and some of their characterizations including ultra uniformization
of ultra approach groups which however have not been considered in [18] although the
idea of ultra approach spaces, and ultra uniform spaces are crept inside in [14] in addition
to some other papers; (b) considering natural connection of neighborhood system with
approach spaces, we characterize approach group by compatible neighborhood system on
group structure; (c) considering approach-Cauchy structures, we show that for a group,
there is a one-to-one correspondence between ultra approach-Cauchy group structures and
strongly normal ultra approach limit group structures, the idea of strong normality first
appeared in [5]; in fact, we prove here that the category of ultra approach-Cauchy groups
and the category of strongly normal ultra approach limit groups are isomorphic.
We arrange these findings as follows. In Section 2, we consider some basic facts that are
used in the sequel. We present the notion of ultra approach groups in Section 3, provide
characterization theorems, and ultra uniformization of ultra approach groups. Using the
notion of neighborhood system as defined in approach space, we give characterization the-
orem on approach group in Section4. Relation between approach groups and approach
limit groups are discussed in Section 5. Finally, we describe the connection between ultra
approach-Cauchy groups and strongly normal approach-Cauchy groups in Section 6.

2. Preliminaries

We denote the set of all filters F,G, ... on a set X by F(X). The point filter of a point
x ∈ X is defined by ẋ = {A ⊆ X : x ∈ A}, or by [x]. The set F(X) is ordered by set
inclusion, i.e., we write F ≤ G if F ⊆ G.

If (Fj)j∈J is a family of filters on a set X, then for a filter U on J , the compressed

operator κ
(
U, (Fj)j∈J

)
is defined by [15]

κ
(
U, (Fj)j∈J

)
=

∨
V ∈U

∧
j∈V Fj .

If F,G ∈ F(X), then the product filter F × G =< {F × G|F ∈ F, G ∈ G} >, i.e., we
have {F ×G|F ∈ F, G ∈ G} as a basis for F×G.

For F,G ∈ F(X) we define



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1764

F⊙G = m (F×G) and F−1 = i(F).

Noting that m (F ×G) = {xy|x ∈ F, y ∈ G} = F ⊙G, we have {F ⊙G|F ∈ F, G ∈ G} as a
basis for F⊙G. Similarly, we find {F−1|F ∈ F} as a basis for F−1, where F−1 = {x−1|x ∈
F}.

Throughout the text for a group (X, ·), we consider e as the identity element.

Lemma 1. Let X and Y be groups, F,G,H ∈ F(X) and f : X −→ Y a group homomor-
phism, then we have
(i) F⊙ F−1 ≤ ė and F−1 ⊙ F ≤ ė;
(ii) ẋ⊙ (ẋ)−1 = (ẋ)−1 ⊙ ẋ = ė;
(iii) ˙̂xy = ẋ⊙ ẏ;

(iv)
˙̂

x−1 = (ẋ)−1;
(v) (F⊙G)⊙H = F⊙ (G⊙H);
(vi) (F−1)−1 = F;
(vii) (F⊙G)−1 = G−1 ⊙ F−1;
(viii) ė⊙ F = F⊙ ė = F;
(ix) (F ∧G)−1 = F−1 ∧G−1;
(x) (F ∧G)⊙H = (F⊙H) ∧ (G⊙H);
(xi) F ≤ ẋ⊙G ⇔ (ẋ)−1 ⊙ F ≤ G (resp. F ≤ G⊙ ẋ ⇔ F⊙ (ẋ)−1 ≤ G) ;
(xii) f(F⊙G) = f(F)⊙ f(G);
(xiii) f(F−1) = (f(F))−1.

A subset Ω ⊂ [0,∞]X is called an ideal in [0,∞]X if for any ξ1, ξ2 ∈ Ω, ξ1 ∨ ξ2 ∈ Ω and
that for any ξ ∈ Ω with ν ≤ ξ implies ν ∈ Ω, where the lattice [0,∞]X is equipped with
the point-wise order.

Definition 1. [15] A collection of ideals Ω = (Ω(x))x∈X in [0,∞]X indexed by the points
of X is called an approach system on X if and only if the following conditions are fulfilled:
(AS1) ∀x ∈ X, ∀ν ∈ Ω(x): ν(x) = 0.
(AS2) ∀x ∈ X,∀ν ∈ [0,∞]X , ∀ϵ > 0, ∀N < ∞, there exists νNϵ ∈ Ω(x) such that ν ∧N ≤
νNϵ + ϵ implies ν ∈ Ω(x).
(AS3) ∀x ∈ X,∀ν ∈ Ω(x), ∀ϵ > 0, N < ∞ there exists (νz) ∈

∏
z∈X Ω(z) such that for any

y, z ∈ X: ν(y) ∧N ≤ νx(z) + νz(y) + ϵ.

For any x ∈ X, ν ∈ Ω(x) is called a local distance in x, and the value ν(t) of a local
distance ν ∈ Ω(x) at a point t ∈ X is interpreted as the distance from x to t according
to ν. Each local distance makes its own measurement of the distance other points in the
space are away from the given point.
A subset B ⊂ [0,∞]X is called a ideal basis in [0,∞]X if for any β1, β2 ∈ B there is a β ∈ B
such that β1 ∨ β2 ≤ β.

Definition 2. [15] A collection of ideal bases B = (B(x))x∈X in [0,∞]X is called an
approach basis if and only if the following statements are true.
(AB1) ∀x ∈ X, ∀β ∈ B(x): β(x) = 0.



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1765

(AB2) ∀x ∈ X, ∀β ∈ B(x), ∀ϵ > 0,∀N < ∞, there exists (βz)z∈X ∈
∏

z∈X B(z) such that
∀y, z ∈ X: β(y) ∧N ≤ βx(z) + βz(y) + ϵ.

Definition 3. [15] If Ω is an approach system, then B = (B(x))x∈X is called a basis for
Ω if and only if the following are satisfied.
(Ab1) ∀x ∈ X, B(x) is a basis for an ideal.

(Ab2) ∀x ∈ X: Ω(x) = B̂(x), where B̂(x) = {ν ∈ [0,∞]X | ∀ϵ > 0,∀N ≤ ∞∃ ξ ∈ B(x) :
ν ∧N ≤ ξ + ϵ}.

Definition 4. [15] A function λ : F(X) −→ [0,∞]X is called a limit operator if and only
if the following conditions are fulfilled:
(AL1) ∀x ∈ X : λ(ẋ)(x) = 0;
(AL2) ∀F,G ∈ F(X), x ∈ X : F ≤ G implies λ(G)(x) ≤ λ(F)(x);
(AL3) ∀ (Fj)j∈J ∈ F(X)J , x ∈ X : λ

(∧
j Fj

)
(x) =

∨
j λ (Fj) (x);

(AL4) ∀G ∈ F(X), (Fy)y∈X ∈ F(X)X , x ∈ X : λ
(
κ
(
G, (Fy)y∈X

))
(x) ≤ λ(G)(x) +∨

y∈X λ (Fy) (y).
Then the pair (X,λ) is called an approach space.

A map f : (X,λ) −→ (Y, λ′) between two approach spaces is called a contraction if
λ′ (f(F)) (f(x)) ≤ λ(F)(x), ∀x ∈ X.

The category of approach spaces and contraction mappings is denoted by AP.

Theorem 1. [15] Let
(
fj : X −→

(
Xj ,Ω

j
)
j∈J

)
be a structured source in AP. Then an

approach basis on X for the unique initial lift of this source in AP is given by

B(x) = {
∨

j∈K νj ◦ fj |K ∈ 2(J),∀j ∈ K : νj ∈ Ωj (fj(x))},∀x ∈ X.

Given an approach space (X,λ), one defines for α ∈ [0,∞] and x ∈ X, the α-
neighborhood filter at x ∈ X given in [13] by

Ux
α =

∧
{F ∈ F(X)|λ(F)(x) ≤ α}.

Remark 1. In an approach space (X,λ), ∀F ∈ F(X), ∀x ∈ X and ∀α ∈ [0,∞]:

λ(F)(x) ≤ α ⇔ F ≥ Ux
α.

[13]

Theorem 2. [13] Let (X,λ) ∈ | AP|. The system U = (Ux
α)x∈X,α∈[0,∞] has the following

properties:
(U0) Ux

α ∈ F(X) for all x ∈ X and α ∈ [0,∞];
(U1) Ux

α ≤ ẋ, for all x ∈ X and α ∈ [0,∞];

(U2) Ux
α+β ≤ κ

(
Ux
β, ((U

y
α)y∈X)

)
, for all x ∈ X and α, β ∈ [0,∞];

(U3) 0 ≤ α ≤ β implies Ux
β ≤ Ux

α;
(U4) For all ∅ ≠ A ⊂ [0,∞]:

∨
α∈AUx

α = Ux
∧A.

We call the system U as the corresponding neighborhood system of the approach space
(X,λ).



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1766

If (X, ·) is a group and ν ∈ [0,∞]X , then for any x ∈ X, we write x ⊙ ν : X −→ [0,∞],
y 7→ x⊙ν(y) = ν(xy). Similarly, we also write ν⊙x : X −→ [0,∞], y 7→ ν⊙x(y) = ν(yx).
If ν ∈ [0,∞]X , then ν−1 : X → [0,∞] is defined by ν−1(x) = ν(x−1). If B ⊂ [0,∞]X , put
x⊙ B = {x⊙ ν| ν ∈ B}. Note that ⟨x⊙ B⟩ = x⊙ ⟨B⟩.
Now for the convenience of the reader, we recall some essential categorical terms that are
needed in the sequel, for the details, we refer to [1].
A functor F : C −→ D is a morphism between categories, consists of mappings between
objects of C and objects of D(sometimes we write as |C| to denote the objects of C) and
the mapping between morphisms of C and morphisms of D such that (i) if f : S −→ T ,
then F(f) : C(S) −→ D(T ); (ii) F(f ◦ g) = F(f) ◦ F(g), whenever f ◦ g is defined; (iii)
F(idS) = idF(S). The functor F is called an embedding if it is injective on objects. If E is
a category, then by a concrete category over E, we understand a pair (G,F), where C is a
category and F : G −→ E is a faithful functor.
A construct is a concrete category over SET, the category of sets, and we consider the ob-
jects of a construct as structured set (S, ξ), and morphisms are suitable mappings between
the underlying sets. A construct is called topological if it allows initial constructions, that
is, for any source (fj : S −→ (Sj , ςj))j∈J , there is a unique structure ς on S such that a
mapping g : (T, β) −→ (S, ς) is a morphism if and only if for each j ∈ J the composition
fj ◦ g : (T, β) −→ (Sj , ςj) is a morphism, where (T, β) is a structured set.
A functor F : C −→ D between categories C and D is called an isomorphism if there is a
functor H : D −→ C such that H◦F = idC and F ◦H = idD. Two categories C and D are
said to be isomorphic if there is an isomorphism.

3. Approach groups, characterizations and ultra uniformization

Definition 5. [15] Let X be a set. A family of ideals (Ω(x))x∈X in [0,∞]X is called an
ultra approach system on X if and only if for all x ∈ X, the following properties are
satisfied:
(AS1) ∀ν ∈ Ω(x): ν(x) = 0.
(AS2) ∀ν ∈ [0,∞]X : (∀ϵ > 0,∀N < ∞,∃νNϵ ∈ Ω(x) s.t. ν∧N ≤ νNϵ +ϵ) implies ν ∈ Ω(x).
(AS3) ∀ν ∈ Ω(x), ∀ϵ > 0, ∀N < ∞,∃ a family (Ω(z))z∈X such that νz ∈ Ωz, ∀z ∈ X and
that ∀y ∈ X, ν(y) ∧N ≤ νx(z) ∨ νz(y) + ϵ.

Definition 6. [15] If (X,Ω) and (X ′,Ω′) are approach spaces (resp.ultra approach spaces),
then a map f : (X,Ω) → (X ′,Ω′) is called contracting at x ∈ X if and only if for all
ν ′ ∈ Ω′(f(x)), for all ϵ > 0 and for all N < ∞ there exists a ν ∈ Ω(x) such that
(ν ′ ◦ f) ∧N ≤ ν + ϵ.
The map f is called a contraction if and only if it is contracting in each x ∈ X.

Definition 7. [14] A family of ideals Ξ in [0,∞]X×X is called an ultra approach unifor-
mity on X if and only if the following properties are satisfied:
(uAU1) ∀ξ ∈ Ξ,∀x ∈ X: ξ(x, x) = 0.
(uAU2) ∀ξ ∈ [0,∞]X×X : (∀ϵ > 0,∀N < ∞,∃ξNϵ ∈ Ξ s.t. ξ ∧N ≤ ξNϵ + ϵ) implies ξ ∈ Ξ.
(uAU3) ∀ξ ∈ Ξ: ξs ∈ Ξ, where ξs(x, y) = ξ(y, x), ∀(x, y) ∈ X ×X.



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1767

(uAU4) ∀ξ ∈ Ξ, ∀ϵ > 0,∀N ≤ ∞,∃ξN ∈ Ξ s.t. ∀x, y, z ∈ X: ξ(x, z) ∧ N ≤ ξN (x, y) ∨
ξN (y, z) + ϵ.

If (X,Ξ) is an ultra approach uniform space, then the underlying approach system of Ξ is
given by Ω(x) = {ξ(x, .)|ξ ∈ Ξ}, or by Ωx.

Definition 8. Let (X, ·) be a group, and Ω = (Ω(x))x∈X be a family of ultra approach
system on X. Then the triple

(
X, ·,Ω = (Ω(x))x∈X

)
is called an ultra approach group if

and only if the following conditions are fulfilled:
(UAG1) the mapping m : X ×X → X, (x, y) 7→ xy is a contraction;
(UAG2) the inversion ȷ : X → X,x 7→ x−1 is a contraction.

Lemma 2. If
(
X, ·,Ω = (Ω(x))x∈X

)
is an approach group, then for any a ∈ X, La the left

(resp. Ra the right) translation is a homeomorphism. That is, bijective and bi-contraction.

Proof. This goes almost in the same way as in the proof of Proposition 2.3 [18] with
La : X → X, x 7→ ax.

Proposition 1. Let
(
X, ·,Ω = (Ω(x))x∈X

)
be an approach group. Then for any x ∈ X,

Ω(x) = {ν ◦ Lx−1 |ν ∈ Ω(e)} (respectively, Ω(x) = {ν ◦ Rx−1 |ν ∈ Ω(e)}).

Proof. Let x ∈ X. If ν ∈ Ω(e), then ν ∈ Ω (Lx−1(x)). Since Lx−1 is a contraction,
by definition, we have ν ◦ Lx−1 ∈ Ω(x). Conversely, if ν ◦ Lx−1 ∈ Ω(x), then ν ◦ Lx−1 ∈
Ω (Lx(e)), which by definition of contraction yields that (ν ◦ Lx−1) ◦ Lx ∈ Ω(e). But
(ν ◦ Lx−1) ◦ Lx = ν, and hence ν ∈ Ω(e). This shows that ν ∈ Ω(e) ⇔ ν ◦ Lx−1 ∈ Ω(x).
The other part follows exactly the same way.

Theorem 3. Let (X, ·) be a group and Ω = (Ω(x))x∈X be an approach system on X.
Then the triple

(
X, ·,Ω = (Ω(x))x∈X

)
is an approach group if and only if the following are

fulfilled:
(a) ∀x ∈ X: Ω(x) = {x−1 ⊙ ν|ν ∈ Ω(e)}, where x−1 ⊙ ν = ν ◦ Lx−1 (alternatively,
Ω(x) = {ν ⊙ x−1| ν ∈ Ω(e)}, where ν ⊙ x−1 = ν ◦ Rx−1);
(b) ∀ν ∈ Ω(e), ∀ϵ > 0, ∀N < ∞, there exists µ ∈ Ω(e), ν−1 ∧ N ≤ µ + ϵ, i.e. ȷ : X →
X,x 7→ x−1 is contracting at e;
(c) ∀ν ∈ Ω(e),∀ϵ > 0,∀N < ∞, there exists µ ∈ Ω(e) such that
∀x, y ∈ X: ν(xy)∧N ≤ µ(x)∨µ(y)+ϵ, i.e. m : (x, y) 7→ xy is contracting at (e, e) ∈ X×X;
(d) ∀ν ∈ Ω(e), ∀ϵ > 0, ∀N < ∞, ∀x ∈ X, there exists µ ∈ Ω(e) such that

(
x⊙ ν ⊙ x−1

)
∧

N ≤ µ+ ϵ, i.e. Intx : z 7→ xzx−1 is contracting at e.

Proof. If
(
X, ·,Ω = (Ω(x))x∈X

)
is an approach group, then (a) follows from the Propo-

sition 1, and (b) follows from the Definition 3.1 [15], while (c) follows the Definition. As for
(d), we employ the Theorem 3.4(b) [15] since Intx = Lx ◦Rx−1 , and both the translations
are contraction maps. To show the converse, assume that (a)-(d) are true. First, we show
that the inversion map ȷ : X → X,x 7→ x−1 is a contraction. Let x ∈ X, ν ∈ Ω (ȷ(x)),



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1768

ϵ > 0, and N < ∞, we want to find a θ ∈ Ω(x) (say) such that (ν ◦ ȷ) ∧N ≤ θ + ϵ. Since
ν ∈ Ω (ȷ(x)), there exists a µ ∈ Ω(e) such that ν = x⊙µ. Consequently, due to axiom (b),
there exists a µ1 ∈ Ω(e) such that µ−1∧N ≤ µ1+ ϵ. Then for any z ∈ X, (ν ◦ ȷ) (z)∧N =
[(x⊙ µ) ◦ ȷ](z)∧N = µ(xz−1)∧N = µ−1

(
zx−1

)
∧N ≤ µ1(zx

−1) + ϵ =
(
µ1 ⊙ x−1

)
(z) + ϵ

⇒ (ν ◦ ȷ) ∧N ≤ θ + ϵ, with θ := µ1 ⊙ x−1, θ ∈ Ω(x). Thus we have proved that for any
ν ∈ Ω (ȷ(x)), ∀ϵ > 0, ∀N < ∞, there exists a θ ∈ Ω(x) such that (ν ◦ ȷ)∧N ≤ θ+ϵ, that is,
ȷ : X → X,x 7→ x−1 is contracting at x, and hence it is contracting in each x, and so, the
inversion ȷ : X → X is a contraction. To prove that the map m : (x, y) 7→ xy is contacting
in (a, b) ∈ X ×X, we employ axioms (a)-(d) in conjunction with Theorem 3.4 [15] to the
contracting maps m,La−1 , Lb−1 ,La and Intb which are respectively contracting at (e, e),
a, b, and e, to get the compositions: m(a, b) = [La ◦ Intb ◦m ◦ (La−1 × Lb−1)](a, b) = ab.
Thus, we have the m : X ×X → X, (x, y) 7→ xy is a contraction map.

Theorem 4. Let (X, ·) be a group, and B be a family of ideals in [0,∞]X such that the
following are fulfilled:
(1) B is an ideal basis, such that ∀ ν ∈ B: ν(e) = 0;
(2) ∀ν ∈ B, ∀ϵ > 0, ∀N < ∞, there exists µ ∈ B such that ν−1 ∧N ≤ µ+ ϵ;
(3) ∀ν ∈ B, ∀ϵ > 0, ∀N < ∞, there exists µ ∈ B such that
∀x, y ∈ X: ν(xy) ∧N ≤ µ(x) ∨ µ(y) + ϵ;
(4) ∀ν ∈ B, ∀ϵ > 0, ∀N < ∞, ∀x ∈ X, there exists µ ∈ B such that(
x⊙ ν ⊙ x−1

)
∧N ≤ µ+ ϵ.

Then there exists a unique approach system such that B is a basis for the approach system
at e and compatible with group structure of X. This approach system is given by: A(x) =
⟨{x−1 ⊙ ν| ν ∈ B}⟩ = ⟨{ν ⊙ x−1| ν ∈ B}⟩.

Proof. In view of the preceding theorem, we only prove (AS3), for this we proceed as
follows.
Let ξ = x−1 ⊙ ν ∈ A(x) with ν ∈ B, let ϵ > 0, and N < ∞. Choose η ∈ B such that
ν(xy) ∧ N ≤ η(x) ∨ η(y) + ϵ. If ξz = z−1 ⊙ η, then ξ(y) ∧ N =

(
x−1 ⊙ ν

)
(y) ∧ N =

ν(x−1y) ∧N = ν
(
x−1zz−1y

)
∧N ≤ η(x−1z) ∨ η(z−1y) + ϵ = ξx(z) ∨ ξz(y) + ϵ.

Proposition 2. Every ultra approach group is ultra approach uniformizable.

Proof. Let (X, ·,Ω) be an ultra approach group. Define
νl : X ×X → [0,∞], (x, y) 7→ νl(x, y) = ν(x−1y)
and
Γ =< {νl ∈ [0,∞]X×X | ν ∈ Ωe} > .
(uAU1) Let x ∈ X and γ ∈ Γ. Then there is a ν ∈ Ωe such that γ(x, x) = νl(x, x) =
ν(e) = 0.
(uAU3) Let ϵ > 0 and γ ∈ Γ. Then there exists ν ∈ Ωe such that γ(x, y) = νl(x, y) =
ν(x−1y). Since by contraction of r, one obtains νl ◦ r ∈ Ωe, yields that ν−1 ∈ Ωe. Thus,
we have γs(x, y) = γ(y, x) = νl(y, x) = ν−1(x−1y) = (ν−1)l(x, y). So, γ

s ∈ Γ.
(uAU4) Let γ ∈ Γ be such that γ(x, y) = νl(x, y) = ν(x−1y). Then for each ϵ > 0
and N < ∞, there exists νNϵ ∈ Ωe such that ν(xy) ∧ N ≤ νNϵ (x) ∨ νNϵ (y). If we put



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1769

γNϵ (x, y) = νNϵ (x−1y), then
γ(x, y) ∧N = ν(x−1y) ∧N
= ν

(
x−1zz−1y

)
∧N

≤ νNϵ (x−1z) ∨ νNϵ (z−1y)
≤ γNϵ (x, z) ∨ γNϵ (z, y) + ϵ.
The underlying ultra-approach structure of Γ is given by
Ω′
x = {γ(x, .)|γ ∈ γ}

= {γ(., x)|γ ∈ Γ}
= {ν ◦ Lx−1 |ν ∈ Ωe}.
In fact, γ(x, .)(y) = γ(x, y) = ν(x−1y) = ν ◦ Lx−1(y), for any y ∈ X. Thus by Proposition
1, we have Ω′

x = Ωx.

4. Characterization of approach groups by neighborhood systems

Definition 9. Let (X, ·) be a group, (X,λ) be an approach space, and U = (Ux
α)x∈X,α∈[0,∞]

be a corresponding neighborhood system of the approach space (X,λ). Then the triple(
X, ·,U = (Ux

α)x∈X,α∈[0,∞]

)
is called a neighborhood approach group if and only if the

following are fulfilled:
(NAGM) Uxy

α∨β ≤ Ux
α ⊙ Uy

β, ∀x, y ∈ X and ∀α, β ∈ [0,∞].

(NAGI) Ux−1

α ≤ (Ux
α)

−1 .

Theorem 5. Let (X, ·) be a group and U = (Ux
α)α∈[0,∞],x∈X be the neighborhood approach

system corresponding to approach space (X,λ). Then
(
X, ·,U = (Ux

α)α∈[0,∞],x∈X

)
is a

neighborhood approach group if and only if the following axioms are fulfilled.
(1) Ue

α ∈ F(X), ∀α ∈ [0, 1];
(2) Ue

α ≤ ė, ∀α ∈ [0,∞];

(3) Ue
α+β ≤ κ

(
Ue
β, (U

y
α)y∈X

)
,∀α, β ∈ [0,∞];

(4) if 0 ≤ α ≤ β, then Ue
β ≤ Ue

α;
(5) Ue

α =
∨

α<β Ue
β;

(6) Ue
α∨β ≤ Ue

α ⊙ Ue
β, ∀α, β ∈ [0,∞];

(7) Ue
α ≤ (Ue

α)
−1 ;

(8) ∀α ∈ [0, 1], ∀x ∈ X: Ux
α = ẋ⊙ Ue

α = Ue
α ⊙ ẋ.

Proof. Let
(
X, ·,U = (Ux

α)α∈[0,∞],x∈X

)
be a neighborhood approach group. Then

conditions (1)-(7) follow immediately. We prove only (8). Since ẋ ≥ Ux
α, we have ẋ ⊙

Ue
α ≥ Ux

α ⊙ Ue
α ≥ Uxe

α∨α = Ux
α. Next, we have: Ux

α = ė ⊙ Ux
α =

(
ẋ⊙ (ẋ)−1

)
⊙ Ux

α =

(ẋ)⊙
(
(ẋ)−1 ⊙ Ux

α

)
≥ ẋ⊙

(
(Ux

α)
−1 ⊙ Ux

α

)
≥ ẋ⊙

(
Ux−1

α ⊙ Ux
α

)
≥ ẋ⊙Ux−1x

α∨α = ẋ⊙Ue
α. This

ends the proof that Ux
α = ẋ ⊙ Ue

α. Similarly, one can obtain the right part. Hence the
results follows.

Conversely, assume that all the conditions (1)-(8) are true.

We need to show that
(
X, ·,U = (Ux

α)α∈[0,∞],x∈X

)
is a neighborhood approach group. As



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1770

it is already a neighborhood approach space, we first prove the condition (NAGM). Let
α, β ∈ [0,∞] and x, y ∈ X. Then by using Lemma 1 repeatedly we get:
Uxy
α∨β = ˙̂xy ⊙ Ue

α∨β
= (ẋ⊙ ẏ)⊙ Ue

α∨β

= ẋ⊙
(
ẏ ⊙ Ue

α∨β

)
≤ ẋ⊙

(
ẏ ⊙

(
Ue
α ⊙ Ue

β

))
(by (6))

= ẋ⊙
(
(ẏ ⊙ Ue

α)⊙ Ue
β

)
= ẋ⊙

(
(Ue

α ⊙ ẏ)⊙ Ue
β

)
(applying (8))

= (ẋ⊙ Ue
α)⊙

(
ẏ ⊙ Ue

β

)
= Ux

α ⊙ Uy
β (again by applying (8)).

To prove (NAGI), note that by applying (7), (8) and Lemma 1, we have:
(Ux

α)
−1 = (ẋ⊙ Ue

α)
−1 = (Ue

α)
−1 ⊙ (ẋ)−1 ≥ Ue

α ⊙ (ẋ)−1 = Ux−1

α .

5. Ultra approach limit group and its relationship with neighborhood
approach group

Definition 10. [4] Let (X, ·) be a group and (X,λ) be an ultra approach limit space. We
call the triple (X, ·, λ) an ultra-approach limit group if the following axioms are satisfied:
(uALM) ∀F,G ∈ F(X), x, y ∈ X : λ(F⊙G)(xy) ≤ λ(F)(x) ∨ λ(G)(y).
(uALI) ∀F ∈ F(X), x ∈ X : λ(F−1)(x−1) ≤ λ(F)(x).

One can notice from [4] that the conditions (uALM) and (uALI) can be replaced by a
single condition, i.e., for all F,G ∈ F(X), x, y ∈ X : λ(F⊙G)(xy−1) ≤ λ(F)(x) ∨ λ(G)(y).

Theorem 6. If (X, ·, λ) is an ultra approach limit group, then(
X, ·,Uλ = (Ux

α)α∈[0,∞],x∈X

)
is a neighborhood approach group, where

Ux
α =

∧
{F ∈ F(X) |λ(F)(x) ≤ α}, for any α ∈ [0,∞] and x ∈ X.

Conversely, if
(
X, ·,U = (Ux

α)α∈[0,∞],x∈X

)
is a neighborhood approach group, then (X, ·, λU )

is an ultra-approach limit group, where λU (F)(x) =
∧
{α ∈ [0,∞] |Ux

α ≤ F}, for any
F ∈ F(X), and x ∈ X.

Proof. Let (X, ·, λ) be an ultra approach limit group. For α, β ∈ [0,∞], and x, y ∈ X,
we put F = Ux

α and G = Uy
β. Then

λ
(
Ux
α ⊙ Uy

β

)
(xy)

= λ (F⊙G) (xy)
≤ λ(F)(x) ∨ λ(G)(y) (by (uALM))
≤ α ∨ β.

This implies that λ
(
Ux
α ⊙ Uy

β

)
(xy) ≤ α ∨ β which in view of Remark 1 yields that Ux

α ⊙



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1771

Uy
β ≥ Uxy

α∨β, i.e., the condition (NAGM) is proved. The condition (NAGI) is an immediate
consequence of Lemma 3.7 [13].
Conversely, in view of Lemma 3.4[13], we only show condition (uALM). Assume that
(NAGM) is true. Let F,G ∈ F(X), and x, y ∈ X. Then
λU (F)(x) ∨ λU (G)(y)

= (
∧
{α ∈ [0,∞] |Ux

α ≤ F}) ∨
(∧

{β ∈ [0,∞]|Uy
β ≤ G}

)
=

∧
{α ∨ β ∈ [0,∞] |Ux

α ≤ F,Uy
β ≤ G} (by using Lemma 2.8[4])

≥
∧
{α ∨ β ∈ [0,∞] |Uxy

α∨β ≤ F⊙G} (as because Uxy
α∨β ≤ Ux

α ⊙ Uy
β ≤ F⊙G, and with the

assumption that F⊙G exists, so is Ux
α ⊙ Uy

β by using Lemma 4.2[4])
= λU (F⊙G) (xy), showing the condition (uALM) is proved. The last condition follows
immediately by using (NAGI) coupled with Lemma 3.7[13].

Corollary 1. Let
(
X, ·,U = (Ux

α)α∈[0,∞],x∈X

)
be a neighborhood approach group. Then

for any α ∈ [0,∞] and x ∈ X: Ux
α = ẋ⊙ Ue

α = Ue
α ⊙ ẋ.

Proof. Let x ∈ X and α ∈ [0,∞]. Then in view of the preceding theorem, Lemma 1
and Lemma 3.8[4], we have
Ux
α =

∧
{F ∈ F(X) |λ(F)(x) ≤ α}

=
∧
{F ∈ F(X) |λ

(
(ẋ)−1 ⊙ F

)
(e) ≤ α}

=
∧
{F ∈ F(X) | (ẋ)−1 ⊙ F ≥ Ue

α} (by Remark 1)
=

∧
{F ∈ F(X) |F ≥ ẋ⊙ Ue

α}
= ẋ⊙ Ue

α.
Similarly, one can show that Ux

α = Ue
α ⊙ ẋ.

If we denote uApLimGrp as the category of all ultra-approach limit groups andNAp-
Grp, the category of neighborhood approach groups associated with approach spaces,
then if follows from [13] in conjunction with the Lemma 3.7[13] and the Theorem 6 above,
these two categories are isomorphic, we leave details for the interested reader. However,
the functors in question, say for instance, F and G are connected as described below:

F :


uApLimGrp −→ NApGrp

(X, ·, λ) 7−→ (X, ·,Uλ)
f 7−→ f

and

G :


NApGrp −→ uApLimGrp
(X, ·,U) 7−→ (X, ·, λU )

f 7−→ f

6. Ultra approach-Cauchy groups, Ultra approach limit groups and
Strongly normal ultra approach limit

In the light of the Section 5[4], we add some results in this section; specifically, our main
aim here is to show that the categories uApChyGrp (the category of ultra approach-



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1772

Cauchy groups) and SNuApLimGrp (the category of strongly normal ultra approach
limit groups) are isomorphic.

Definition 11. [17] Let X be a set. A mapping Υ: F(X) −→ [0,∞] is called an approach-
Cauchy structure if and only if the following conditions are fulfilled:
(AChy1) Υ(ẋ) = 0;
(AChy2) F ≤ G implies Υ(G) ≤ Υ(F) for all F,G ∈ F(X);
(AChy3) Υ(F ∧G) ≤ Υ(F) + Υ(G).
Then the pair (X,Υ) is called an approach-Cauchy space.
A mapping f : (X,Υ) −→ (X ′,Υ′) between approach-Cauchy spaces is called approach-
Cauchy contraction if and only if for all F ∈ F(X), Υ′(f(F)) ≤ Υ(F).
The category of all approach-Cauchy spaces and approach-Cauchy contractions is denoted
by ApChy.

Definition 12. Let (X, ·) be a group and (X,Υ) be an approach-Cauchy space. Then the
triple (X, ·,Υ) is called an approach-Cauchy group if and only if for all F,G ∈ F(X):
Υ(F⊙G−1) ≤ Υ(F) + Υ(G).
The category of all approach-Cauchy groups and approach-Cauchy contractions which are
homomorphisms denoted by ApChyGrp.

Definition 13. [4, 17] A map Υ : F(X) → [0,∞] is called an ultra approach-Cauchy
structure on X if and only if the following axioms are fulfilled:
(uAChy1) ∀x ∈ X: Υ(ẋ) = 0;
(uAChy2) ∀F,G ∈ F(X) with F ≤ G, Υ(G) ≤ Υ(F),
(uAChy3) ∀F,G ∈ F(X), if F ∨G exists, then Υ(F ∩G) ≤ Υ(F) ∨Υ(G).
A mapping f : (X,Υ) −→ (Y,Υ′) between ultra approach-Cauchy spaces is called Ultra
approach-Cauchy contraction or Cauchy contraction if and only if for all F ∈ F(X),
Υ′(f(F)) ≤ Υ(F).
The category of all ultra approach-Cauchy spaces and contractions is denoted by uApChy.

Definition 14. Let Υ : F(X) → [0,∞] be an ultra approach-Cauchy structure on a group
(X, ·), then the triple (X, ·,Υ) is called an ultra approach-Cauchy group if and only if the
mapping h : (X ×X,Υ×Υ) → (X,Υ) , (x, y) 7→ x−1y is a contraction, (or, equivalently,
∀F,G ∈ F(X), Υ

(
F⊙G−1

)
≤ Υ(F) ∨Υ(G).)

Note that ∀F ∈ F(X), Υ(F−1) ≤ Υ(F), and ∀F,G ∈ F(X), Υ(F⊙G) ≤ Υ(F)∨Υ(G), are
also hold good.
The category of ultra approach-Cauchy groups and Cauchy contractive homomorphisms is
denoted by uApChyGrp.

Proposition 3. uApChyGrp is a topological category.

Proof. Let (X, ·) be a group, fj : X −→ Xj a group homomorphism, and (Xj , ·, (Υj)j∈J)
be a family of ultra approach-Cauchy groups. Consider a source S = (fj : X −→ (Xj , ·,Υj))j∈J .
Then in view of the Section 5[17] the ultra approach-Cauchy structure on X is given for
any F ∈ F(X), by Υ(F) =

∨
j∈J Υj(f(j(F))). It is proved in [17] that the source S has



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1773

initial structure and that the pair (X,Υ) is an ultra approach-Cauchy space. It remains
to be checked the following single condition:
Thus, for any F,G ∈ F(X), we have
Υ
(
F⊙G−1

)
=

∨
j∈J Υj∈J

(
fj(F⊙G−1)

)
=

∨
j∈J Υj

(
fj(F)⊙ fj(G)−1

)
≤

∨
Υj∈J(fj(F)) ∨

∨
j∈J Υj(fj(G)) = Υ(F) ∨Υ(G), for all j ∈ J .

Finally, we show that for any ultra approach-Cauchy group (Y, ·,Υ′), a group homomor-
phism g : (Y, ·,Υ′) −→ (X, ·,Υ) is a contraction if and only if for all j ∈ J , the map
fj ◦ g : (Y, ·,Υ′) −→ (Xj , ·,Υj) is a contraction. But the composition fj ◦ g is clearly
contraction, while the contraction of g follows at once from [17].

Recall that an ultra approach limit group, [3], is a triple (X, ·, λ) consisting of a group
(X, ·) and an ultra approach limit structure λ : F(X) → [0,∞]X meaning for all x ∈ X,
λ(ẋ) = 0, for all F,G ∈ F(X) with F ≤ G implies λ(G) ≤ λ(F) and λ(F∧G) = λ(F)∨λ(G),
such that the group operation h : X × X → X, (x, y) 7→ xy−1 is a contraction, i.e.,
∀F,G ∈ F(X), λ

(
F⊙G−1

)
(xy−1) ≤ λ(F)(x) ∨ λ(G)(y). With each ultra approach limit

group (X, ·, λ), there is associated a natural ultra approach-Cauchy structure defined as
follows:
Υλ : F(X) → [0,∞],F 7→ Υλ(F) = λ

(
F−1 ⊙ F

)
(e) ∨ λ

(
F⊙ F−1

)
(e).

On the other hand, every ultra approach Cauchy group (X, ·,Υ) gives rise to an ultra
approach limit structure given by:
λΥ : F(X) → [0,∞]X , F 7→ λΥ(F)(x) = Υ (F ∩ ẋ).

Lemma 3. [4] Let (X, ·, λ) be an ultra approach limit group, F ∈ F(X) and x ∈ X. Then
λ(F)(x) = λ([x]−1 ⊙ F)(e) = λ(F⊙ [x]−1)(e).

Proposition 4. If (X, ·, λ) is an ultra approach limit group, then
Υλ : F(X) → [0,∞] defined by
Υλ(F) = λ

(
F−1 ⊙ F

)
(e) ∨ λ

(
F⊙ F−1

)
(e), ∀F ∈ F(X)

gives rise to an ultra approach-Cauchy structure.

Proof. (uAChy1) For any x ∈ X, Υλ(ẋ) = λ
(
ẋ−1 ⊙ ẋ

)
(e)∨λ

(
ẋ⊙ ẋ−1

)
(e) = λ(ė)(e)∨

λ(ė)(e) = 0.
(uAChy2) If F ≤ G, then since F−1 ⊙ F ≤ G−1 ⊙G, we have Υλ(G) ≤ Υλ(F).
(uAChy3) Let F,G ∈ F(X) such that F∨G exists, then F−1⊙G ≤ ė and also, G−1⊙F ≤ ė,
hence upon using these we have
Υλ(F) ∨Υλ(G)
= λ

(
F−1 ⊙ F

)
(e) ∨ λ

(
F⊙ F−1

)
(e) ∨

(
G−1 ⊙G

)
(e) ∨ λ

(
G⊙G−1

)
(e)

= [λ
(
F−1 ⊙ F

)
(e) ∨

(
G−1 ⊙G

)
(e)]

∨
[λ

(
F⊙ F−1

)
(e) ∨ λ

(
G⊙G−1

)
(e)]

≥ λ
(
F−1 ⊙G

)
(e) ∨ λ

(
G−1 ⊙ F

)
(e)

Thus, we have
Υλ (F ∩G)

= λ
(
(F ∩G)−1 ⊙ (F ∩G)

)
(e) ∨ λ

(
(F ∩G)⊙ (F ∩G)−1

)
(e)

= λ
((
F−1 ⊙ F

)
∩
(
F−1 ⊙G

)
∩
(
G−1 ⊙ F

)
∩
(
G−1 ⊙G

))
(e)

∨
λ
((
F⊙ F−1

)
∩
(
F⊙G−1

)
∩
(
G⊙ F−1

)
∩
(
G⊙G−1

))
(e)



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1774

= [λ
(
F−1 ⊙ F

)
(e) ∨ λ

(
F⊙ F−1

)
(e)]

∨
[λ

(
G−1 ⊙G

)
(e) ∨ λ

(
G⊙G−1

)
(e)]∨

[λ
(
F−1 ⊙G

)
(e) ∨ λ

(
G⊙ F−1

)
(e)]

∨
[λ

(
G−1 ⊙ F

)
(e) ∨ λ

(
F⊙G−1

)
(e)]

≤ Υλ(F) ∨Υλ(G) ∨Υλ(G) ∨Υλ(F) = Υλ(F) ∨Υλ(G).

Definition 15. An ultra approach limit group (X, ·, λ) is called strongly normal if and only
if for all F,G ∈ F(X), λ

(
F⊙G⊙ F−1

)
(e) ≤ λ

(
F⊙ F−1

)
(e) ∨ λ

(
F−1 ⊙ F

)
(e) ∨ λ(G)(e).

Proposition 5. Let (X, ·, λ), and (X ′, ·, λ′) be strongly normal ultra approach limit groups.
If f : X −→ X ′ is a group homomorphism, then the following assertions are equivalent:
(i) f : (X,λ) −→ (X ′, λ′) is a contraction;
(ii) f : (X,Υλ) −→ (X ′,Υλ′) is Cauchy contraction.

Proof. Assume (i) holds. Then using Lemma 1, for any F ∈ F(X),
Υλ′(f(F)) = λ′ (f(F)−1 ⊙ f(F)

)
(f(e))∨λ′ (f(F)⊙ f(F)−1)

)
(f(e)) = λ′ (f(F−1 ⊙ F)

)
(f(e))∨

λ′ (f(F⊙ F−1)
)
(f(e)) ≤ λ(F−1 ⊙ F)(e)∨ λ(F⊙ F−1)(e) = Υλ(F), i.e., Υλ′(f(F)) ≤ Υλ(F).

Now assume (ii), and let F ∈ F(X), and x ∈ X. Then upon using Lemma 3, we get
λ′(f(F))(f(x)) = λ′ ([f(x)]−1 ⊙ f(F)

)
(f(e)) ∨ λ′ (f(F)⊙ [f(x)]−1

)
(f(e))

≤ λ′ (([f(x)] ∧ f(F ))−1 ⊙ ([f(x)] ∧ f(F)
)
(f(e))∨λ′ (f(F) ∧ [f(x)]⊙ (f(F) ∧ [f(x)])−1

)
(f(e))

= Υλ′(f(F)∧[f(x)]) ≤ Υλ(F∧[x]) = λ((F∧[x])−1⊙(F∧[x])(e)∨λ((F∧[x])⊙(F∧[x])−1)(e) ≤
λ(F−1⊙F)(e = x−1x)∨λ(F⊙F)(e = xx−1) ≤ λ(F)(x)∨λ(F−1)(x−1) ≤ λ(F)(x)∨λ(F)(x) =
λ(F)(x), which proves that λ′(f(F))(f(x)) ≤ λ(F)(x), i.e., f : X −→ X ′ is a contraction.

Lemma 4. If (X, ·,Υ) is a ultra approach-Cauchy group, then (X, ·, λΥ) is an ultra
approach-limit group.

Proof. In view of the Proposition 5.16 [17], we only need to show that the group
operation h : X ×X → X, (x, y) 7→ xy−1 is a contraction. If F,G ∈ F(X) and x, y ∈ X,
then by the Lemma 1(iii), we have(
F⊙G−1

)
∩

˙̂
xy−1 =

(
F⊙G−1

)
∩
(
ẋ⊙ ẏ−1

)
≥ (F ∩ ẋ)⊙

(
G−1 ∩ ẏ−1

)
.

Upon using (uAChy2), we get
Υ
((
F⊙G−1

)
∩
(
ẋ⊙ ẏ−1

))
≤ Υ

(
(F ∩ ẋ)⊙

(
G−1 ∩ ẏ−1

))
= Υ

(
(F ∩ ẋ)⊙ (G ∩ ẏ)−1

)
≤ Υ(F ∩ ẋ) ∨Υ(G ∩ ẏ)
= λΥ(F)(x) ∨ λ(G)(y).
Consequently, we have

λΥ

(
F⊙G−1

)
(xy−1) = Υ

((
F⊙G−1

)
∩

˙̂
xy−1

)
≤ λΥ(F)(x) ∨ λΥ(G)(y).

Lemma 5. If (X, ·, λ) is an ultra approach limit group, then (X, ·,Υλ) is an ultra approach-
Cauchy group if and only if it is strongly normal approach limit groups.

Proof. Define Υ(F) = λ(F−1 ⊙ F)(e) ∨ λ(F⊙ F−1)(e) ∨ λ(G)(e), for all F,G ∈ F(X)
Assume that (X, ·, λ) be a strongly normal approach limit group, we prove that (X, ·,Υλ)



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1775

is an ultra approach-Cauchy group.
In view of the Proposition 4, only we need to prove that the group operations are contrac-
tive. Since (X, ·, λ) is strongly normal, we have for any F,G ∈ F(X),
Υ(F⊙G) = λ((F⊙G)⊙ (F⊙G)−1)(e) ∨ λ((F⊙G)−1 ⊙ (F⊙G))(e)
= λ

(
F⊙G⊙G−1 ⊙ F−1

)
(e)∨λ(G−1⊙F−1⊙F⊙G)(e) ≤ λ(F−1⊙F)(e)∨λ(F⊙F−1)(e)∨

λ(G ⊙ G−1)(e) ∨ λ((G ⊙ G))(e) ∨ λ(G−1 ⊙ G−1))(e) ∨ λ(F−1 ⊙ F)(e) (by applying λ to
[e] = [e]⊙ [e] ≤ G⊙G)
≤ λ(F−1⊙F)(e)∨λ(F⊙F−1)(e)∨λ(G⊙G−1)(e)∨λ(G−1⊙G)(e) = Υ(F)∨Υ(G), proving
that Υ(F⊙G) ≤ Υ(F) ∨Υ(G). Finaly, for any F ∈ F(X), we have
Υ(F−1) = λ((F−1)−1 ⊙ F−1)(e) ∨ λ(F−1 ⊙ (F−1)−1)(e)
λ((F ⊙ F−1)(e) ∨ λ(F−1 ⊙ (F)(e) = Υ(F). This ends the prove that (X, ·,Υ) is an ultra
approach-Cauchy group.

Thus, there are two functors A and B which in conjunction with the Proposition 5
yields the following:

A :


uApChyGrp −→ SNApLimGrp

(X, ·,Υ) 7−→ (X, ·, λΥ)
f 7−→ f

and

B :


SNApLimGrp −→ uApChyGrp

(X, ·, λ) 7−→ (X, ·,Υλ)
f 7−→ f

Theorem 7. uApChyGrp is isomorphic to SNApLimGrp.

Proof. Observed that

SNApLimGrp
B−→uApChyGrp

A−→SNApLimGrp:
(X, ·, λ) 7−→ (X, ·,Υλ) 7−→ (X, ·, λ);

then one can check that A ◦B = idSNApLimGrp, i.e., λΥλ
= λ. In fact, for any F ∈ F(X),

and x ∈ X, we have
λΥλ

(F)(x) = Υλ(F ∧ [x]) = λ
(
(F ∧ [x])−1 ⊙ (F ∧ [x])

)
(e) ∨ λ

(
(F ∧ [x])⊙ (F ∧ [x])−1

)
(e)

= λ
(
([x]−1 ∧ F−1)⊙ (F ∧ [x])

)
(e) ∨ λ

(
(F ∧ [x])⊙ (F−1 ∧ [x]−1)

)
(e)

≥ λ
(
([x]−1 ∧ F−1)⊙ (F ∧ [x])

)
(e) ≥ λ(F⊙ [x]−1) = λ(F)(x), this is so, because of the fact

that [x]−1 ∧ F−1)⊙ (F ∧ [x]) ≤ [x]−1 ⊙ F, the applying λ to get λΥλ
≥ λ. Conversely, for

any F ∈ F(X) and x ∈ X, we have
λΥλ

(F)(x) = Υλ(F ∧ [x]) = λ
(
(F ∧ [x])−1 ⊙ (F ∧ [x])

)
∨ λ

(
(F ∧ [x])⊙ (F ∧ [x])−1)

)
= λ

(
(F ∧ [x])−1 ⊙ (F ∧ [x])

)
∨ λ

(
(F ∧ [x])⊙ (F−1 ∧ [x]−1)

)
.

Since in one hand, (F∧ [x])⊙(F−1∧ [x]−1) = (F⊙F−1)∧(F⊙ [x]−1)∧([x]⊙F−1)∧ [x]⊙ [x]−1

and the other (F∧ [x])−1 ⊙ (F∧ [x]) = (F−1 ⊙F)∧ (F−1 ⊙ [x])∧ ([x]−1 ⊙F)∧ ([x]−1 ⊙ [x]),
one obtains
= λ

(
(F ∧ [x])−1 ⊙ (F ∧ [x])

)
∨ λ

(
(F ∧ [x])⊙ (F−1 ∧ [x]−1)

)
≤ λ

(
(F−1 ⊙ F)(e = x−1x) ∨ λ(F−1 ⊙ [x])(e) ∧ λ(F⊙ [x]−1)(x) ∨ λ([e])(e)

)



T.M.G. Ahsanullah, Fawzi Al-Thukair / Eur. J. Pure Appl. Math, 17 (3) (2024), 1762-1778 1776

∨
(
λ(F⊙ F−1)(e = xx−1) ∨ λ(F⊙ [x]−1)(e) ∨ λ(([x] ∧ F−1)(e) ∨ λ[e](e)

)
≤ λ(F)(x).

This is so, because of the fact that λ(F−1⊙F)(x−1x) ≤ λ(F−1)(x−1)∨λ(F(x) ≤ λ(F)(x)∨
λ(F)(x) = λ(F)(x), and continuing in this way, we can do other parts including applying
homogeneity. Thus, we can prove that λΥλ

≤ λ, and hence λΥλ
= λ. For the other

direction, we look at the scheme below:

uApChyGrp
B−→SNApLimGrp

A−→uApChyGrp:
(X, ·,Υ) 7−→ (X, ·, λΥ) 7−→ (X, ·,Υ);

then one can check that B ◦ A = iduApChyGrp, i.e., ΥλΥ
= Υ. In fact, for any F ∈ F(X),

ΥλΥ
(F) = λΥ(F−1 ⊙ F)(e) ∨ λΥ(F⊙ F−1)(e) = Υ(F). Hence the result follows.

Corollary 2. If the underlying group is Abelian, then uApChyGrp is isomorphic to
uApLimGrp.

7. Conclusion

In this paper, from categorical perspective, we considered two isomorphisms, one
between the categories uApLimGrp (the category of all ultra-approach limit groups)
and NApGrp (the category of neighborhood approach groups associated with approach
spaces); another, between the categories uApChyGrp (the category of ultra approach-
Cauchy groups) and SNApLimGrp (the category of strongly normal approach limit
groups) besides discussing some characterizations of (ultra)-approach groups. Since ul-
tra approach structures originated from the idea of non-archimedean structure or ultra
metrics, we are interested to associate all of these structures in relation to ultra approach
metric groups, and the ∞p-metrizability of ultra approach groups. These questions are
yet to be established. We hope to settle these issues in one of our forthcoming papers.
It would be interesting to add some applications like those in [10–12] that are based on
rough sets and their generalizations. Since as this stage we do not see any direct rela-
tionship of our work with rough sets, one of the reasons could be that rough sets are
generalization of Zadeh’s concept of fuzzy sets whereas our findings are based on non-
fuzzy, rather it is based on non-Archimedean metric spaces; however, we will look into
this extraordinary situations in our future research. However, for better understand the
soft sets and their applications in medical science, we refer to some of the papers which
are definitely interesting in their own right such as [10–12]. However, approach spaces
have significant applications within mathematics, such as, functional analysis, and much
beyond which can be find out in [16], where one can also find plenty of examples on ap-
proach spaces, and their connected branches. For further examples on approach groups,
we refer to [4].

Acknowledgements

We are sincerely grateful to the reviewers for their useful comments and suggestions.



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