



































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

Algebraic and Topological Analysis of Enveloping Semigroups in 

Transformation Groups: Proximal Equivalence and Homomorphic 

Image 
 

Michael N. John 

Department of Mathematics, AkwaIbom State University, Nigeria 

 

And 

 

UdoakaOtobong. G. 

Department of Mathematics, AkwaIbom State University, Nigeria 

 

Abstract 

This paper investigates the algebraic properties of the enveloping semigroupE of a 

transformation group (X,T,μ) with a compact Hausdorff phase space X. The transition 

group G is considered as a group of homeomorphisms on X, and E is defined as the 

closure of G in X×X. The main focus is on establishing a connection between the 

proximal equivalence relation in X and the structure of E, particularly the presence of a 

unique minimal right ideal. In the latter part, the study extends to the analysis of 

homomorphic images of transformation groups through their enveloping semigroups. 

KEYWORDS:Algebraic Cryptography, Group Theory, Enveloping Semigroup, Proximal 

Equivalence, Homomorphic Images, Compact Hausdorff Space, Transition Group, 

Minimal Right Ideal. 

 

1. INTRODUCTION 

The study of transformation groups with compact Hausdorff phase spaces has 

significant implications in various mathematical and applied fields. Bowen's [1] 

foundational work explores the concept of proximal equivalence in topological 

dynamics, providing insights into the connection between dynamical systems and the 

relations studied in his paper.[2]and [29], contributed to the study of enveloping 

semigroups in topological transformation groups, offering valuable insights into their 

algebraic properties and role in capturing the dynamics of homeomorphisms. [3] Work 

focuses on minimal right ideals in semigroups arising from continuous maps, providing a 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

relevant perspective for the investigation of such ideals in enveloping semigroups. [4], 

also contributed to the understanding of enveloping semigroups in the context of 

topological dynamics, emphasizing their role in capturing dynamic behavior through 

algebraic structures.This paper focuses on the enveloping semigroupE associated with 

such groups, exploring its algebraic and topological properties. The transition groupG is 

viewed as a group of homeomorphisms, and E is defined as the closure of G in X×X. 

We aim to establish a link between the proximal equivalence relation in X and the 

structure of E, specifically the existence of a unique minimal right ideal. Additionally, we 

delve into the analysis of homomorphic images of transformation groups through their 

enveloping semigroups. See [26], [27] and [31] 

2. PRELIMINARIES 

Transformation Groups 2.1Let's look into the mathematical definition of a transformation 

group (X,T,μ) with a compact Hausdorff phase space X, provide an illustration, and explore an 

example. 

Mathematical Definition: 

1. Compact Hausdorff Phase Space X: 

 X is a topological space that is both compact and Hausdorff. Compactness 

ensures that every open cover has a finite subcover, and Hausdorffness 

guarantees the separation of distinct points by disjoint open sets. 

2. Group of Homeomorphisms T: 

 T is a group consisting of homeomorphisms from X to itself. A 

homeomorphism is a continuous bijective map with a continuous inverse, 

preserving the topological structure of the space. 

3. Continuous Action μ: 

 The action μ:T×X→X represents how elements of the group T act on the 

space X. It is a continuous map satisfying: 

 μ(e,x)=x for all x∈X, where e is the identity element of T. 

 μ(g,μ(h,x))=μ(gh,x) for all g,h∈T and x∈X. 

Illustration 2.2. Consider a transformation group on the unit circle in the complex plane. 

Let X be the unit circle, T be the group of rotations around the circle, and μ be the action 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

of rotating points. Each element in T is a rotation, and the action μ is the composition of 

rotations. The conditions ensure that the identity rotation leaves points unchanged, and 

the composition of rotations is associative. 

Example 2.3. Let X be the interval [0,1], and T be the group of all homeomorphisms of 

[0,1] to itself, such as translations, reflections, and compositions of such 

transformations. The action μ can be the translation of points. For a translation t∈T and 

a point x∈X, the action μ(t,x) represents the new position of the point after the 

translation. The group structure ensures that the identity transformation leaves points 

unchanged, and the composition of transformations is associative. 

In summary, a transformation group with a compact Hausdorff phase space involves a 

topological space, a group of homeomorphisms, and a continuous action representing 

transformations. The example on the unit circle and interval illustrates how such groups 

can capture symmetries and actions on different spaces. 

 

Enveloping Semigroup 2.2.Let (X,G,μ) be a transformation semigroup with a phase space 

X, a semigroupG of transformations on X, and a continuous action μ:G×X→X. The enveloping 

semigroupE is defined as the closure of the transition semigroupG in the product space X×X, 

denoted as � = �̅ 

This closure operation ensures that the product of any two elements in G remains in the 

enveloping semigroup, providing a continuous extension to the transition semigroup. 

Example 2.3. Consider a transition semigroup of rotations G acting on a unit circle X in 

the complex plane. Each element in G is a rotation around the circle. The action μ 

rotates points on the circle. The enveloping semigroupE is the closure of G in the 

product space X×X, where the product of two rotations remains in E. 

Illustration 2.4.Imagine a clock face representing the unit circle, and G as the set of all 

possible hour-hand rotations. If you rotate the hour hand to 3 and then to 4, the resulting 

position lies in the enveloping semigroupE. The closure ensures that the product of any 

two rotations is also a valid rotation, creating a continuous structure on the clock face. 

This extension captures all possible positions of the hour hand under continuous 

rotations, forming the enveloping semigroupE. 

In summary, the enveloping semigroup is a closure of the transition semigroup in the 

product space, providing a continuous extension to the original semigroup. The example 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

of rotations on a unit circle illustrates how this enveloping semigroup captures all 

possible continuous transformations on the given phase space. 

3. DEFINITION OF TERMS 

Proximal Equivalence 3.1.Let (X,G,μ) be a transformation semigroup with a phase space X, 

a semigroupG of transformations on X, and a continuous action μ:G×X→X. The enveloping 

semigroupE is the closure of G in the product space X×X, i.e., � = �̅. Proximal equivalence is 

then a relation∼ on X defined as follows: 

For x,y∈X, we say that x is proximally equivalent to y, denoted x∼y, if there exists a 

sequence (gn)⊆G such that limn→∞μ(gn,x)=limn→∞μ(gn,y). 

In simpler terms, two points x and y are proximally equivalent if there exists a sequence 

of transformations from the enveloping semigroupE such that the images of x and y 

under these transformations converge to the same point. 

Illustration 3.2.Consider a transformation semigroupG consisting of all translations on 

the real line X. The enveloping semigroupE is the closure of G in the product space 

R×R. Proximal equivalence in this context would mean that two points x and y are 

considered equivalent if there exists a sequence of translations that brings x and y 

arbitrarily close to each other. 

Example 3.3. Let X=R, and G be the semigroup of positive translations, i.e., G={ta

∣a>0}, where ta(x)=x+a. The enveloping semigroupE is the closure of G. Two points x 

and y are proximally equivalent if there exists a sequence(an) such that limn→∞(x+an

)=limn→∞(y+an). 

Proximal equivalence is a relation on a phase space X determined by the behavior of 

transformations in the enveloping semigroupE. Two points are considered proximally 

equivalent if there is a sequence of transformations from E that brings them arbitrarily 

close to each other. 

Homomorphic Images 3.4.Let (X,G,μ) be a transformation semigroup with a phase space X, 

a semigroupG of transformations on X, and a continuous action μ:G×X→X. The enveloping 

semigroupE is the closure of G in the product space X×X, i.e., � = �̅. Now, homomorphic 

images can be defined as the images of the transformations in E under a homomorphism 

mapping to another group. 

Let H be a group and ϕ:E→H be a homomorphism such that ϕ(xy)=ϕ(x)ϕ(y) for all 

x,y∈E. The set of homomorphic images is then defined as: 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

Homomorphic Images={ϕ(x)| x∈E} 

This set represents the images of the elements in the enveloping semigroupE under the 

homomorphism ϕ into the group H. 

Illustration 3.5. Consider a transformation semigroupG consisting of all rotations on a 

circle, and let E be its enveloping semigroup. Now, suppose H is the additive group of 

integers, and ϕ:E→H is a homomorphism that assigns each rotation an integer value 

corresponding to the number of degrees rotated. The homomorphic images, in this 

case, would be the set of integers representing the degrees of rotation. 

Example 3.6. Let G be the semigroup of all positive real number transformations on the 

real line X, i.e., G={ta |a>0}, where ta(x)=x+a. The enveloping semigroupE is the closure 

of G. Now, consider the additive group of integers H, and define a homomorphism 

ϕ:E→H such thatϕ(ta)=⌊a⌋, where ⌊a⌋ is the greatest integer less than or equal to a. The 

homomorphic images in this case would be the set of integers corresponding to the floor 

values of the translation parameters. 

Homomorphic images in the context of enveloping semigroups involve mapping 

transformations to another group through a homomorphism. The mathematical definition 

captures this concept, and the illustration and example demonstrate how 

transformations in the enveloping semigroup can be mapped to homomorphic images in 

different groups. 

4. CENTRAL IDEA 

Lemma 4.1. For a transformation group (X,T,μ), where X is a topological space, T is a group of 

homeomorphisms on X, and μ:T×X→X is a continuous action, the enveloping semigroupE is a 

group of homeomorphisms on X. 

Proof: 

1. Closure under Composition: 

 Let f,g∈E. Since E is the closure of T in the product space X×X, there exist 

sequences (tn)⊆T converging to f and (sn)⊆T converging to g. Consider 

the composition f∘g. We need to show that f∘g is also in E. 

 By the continuity of the action μ, we have μ(tn,x)→f(x) and μ(sn,x)→g(x) for 

all x∈X as n→∞. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

 Now, consider μ(tn⋅sn,x). By the group action property, μ(tn⋅sn,x)=μ(tn,μ(sn

,x)). 

 As n→∞,μ(tn⋅sn,x)→f(g(x)) because of the continuity of μ and the 

convergence of (tn) and (sn). 

 Therefore, f∘g is in the closure of T, i.e., f∘g∈E. 

2. Existence of Identity Element: 

 Let e be the identity element of the group T. Since e is a homeomorphism, 

it is also in E as the constant sequence converges to e. 

3. Existence of Inverses: 

 Let f∈E. Since f is in the closure of T, there exists a sequence (tn)⊆T 

converging to f. 

 Consider the sequence (tn
−1), where tn

−1 is the inverse of each tn in T. As T 

is a group, tn
−1 is also in T. 

 The sequence (tn
−1) converges to f−1 because the inverse is a continuous 

operation on T. 

 Therefore, f−1 is in the closure of T, i.e., f−1∈E. 

4. Closure under Topological Composition: 

 The composition of homeomorphisms is itself a homeomorphism. Since T 

consists of homeomorphisms and E is the closure of T, every element of E 

is a homeomorphism. 

Hence, E satisfies the group axioms of closure under composition, the existence of an 

identity element, and the existence of inverses. Therefore, E is a group of 

homeomorphisms on X. 

Proposition 4.2.The proximal equivalence relation in X is an equivalence relation if and only if 

there exists only one minimal right ideal in E. 

Proof. 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

1.Proximal Equivalence as an Equivalence Relation: 

Let ∼ be the proximal equivalence relation on X. We will show that ∼ is an equivalence 

relation. 

 Reflexivity: For any x∈X, x∼x since the sequence of identity transformations in E 

converges to x. 

 Symmetry: If x∼y, then there exists a sequence (gn)⊆E such thatlimn→∞μ(gn

,x)=limn→∞μ(gn,y). Therefore, y∼x as well. 

 Transitivity: If x∼y and y∼z, then there exist sequences (gn) and (hn) in E such 

that limn→∞μ(gn,x)=limn→∞μ(gn,y) and limn→∞μ(hn,y)=limn→∞μ(hn,z). The 

concatenation of these sequences, (gn⋅hn), is also in E by the group properties. 

Furthermore, limn→∞μ(gn⋅hn,x)=limn→∞μ(gn,μ(hn,x))=limn→∞μ(gn,y)=limn→∞μ(hn,z), 

implying x∼z. 

2. Existence of One Minimal Right Ideal in E: 

Now, let's show the converse. Assume there exists only one minimal right ideal in E. We 

need to show that ∼ is an equivalence relation. 

 Reflexivity: By the definition of minimal right ideals, there exists a sequence (gn

)⊆E such that limn→∞μ(gn,x)=x. 

 Symmetry: If x∼y, then there exists a sequence (gn)⊆E such that limn→∞μ(gn

,x)=limn→∞μ(gn,y). Since there is only one minimal right ideal, (gn
−1) is also in E, 

and limn→∞μ(gn
−1,x)=limn→∞μ(gn

−1,μ(gn,x))=limn→∞μ(gn
−1⋅gn,x)=limn→∞μ(e,x)=x. 

Therefore, y∼x. 

 Transitivity: If x∼y and y∼z, there exist sequences (gn) and (hn) in E such that 

limn→∞μ(gn,x)=limn→∞μ(gn,y) and limn→∞μ(hn,y)=limn→∞μ(hn,z).  

The concatenation of these sequences, (gn⋅hn), is also in E by the group 

properties. Furthermore, limn→∞μ(gn⋅hn,x)=limn→∞μ(gn,μ(hn,x))=limn→∞μ(gn

,y)=limn→∞μ(hn,z), implying x∼z. 

Therefore, the proximal equivalence relation is an equivalence relation if and only if 

there exists only one minimal right ideal in E. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

Theorem 4.3.The algebraic structure of E directly correlates with the recursive properties of 

the transformation group T. 

Proof: 

Let (X,T,μ) be a transformation group with a phase space X, a group T of 

transformations on X, and a continuous action μ:T×X→X. The enveloping semigroup is 

denoted as � = ��, the closure of T in the product space X×X. 

Correlation between Algebraic Structure and Recursive Properties: 

1. Algebraic Structure of E: 

 The enveloping semigroupE is a closure of T, encompassing all possible 

compositions and limits of transformations in T. The elements of E are 

sequences of transformations that converge to a limit in X×X. 

2. Recursive Properties of T: 

 The recursive properties of T involve the composition of transformations, 

where each transformation in T maps points in X to other points. The 

recursion represents the repeated application of these transformations. 

Proof of Correlation: 

The algebraic structure of E directly correlates with the recursive properties of T due to 

the closure operation: 

 Composition of Transformations in T: 

 The closure of T in E ensures that the composition of transformations in T 

remains within E. This closure is essential for capturing the recursive 

nature of transformations in T. 

 Limits and Convergence: 

 The closure operation allows the inclusion of limit points in E. As 

transformations in T are composed and iterated, the limits of these 

compositions, if they exist, are captured in E. This reflects the recursive 

behavior of T as transformations are applied repeatedly. 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

 Topological Structure: 

 The topological closure ensures that E captures not only the algebraic 

composition of transformations but also the continuity and convergence 

properties. This is crucial for understanding the recursive nature of 

transformations in T within the topological space X. 

Therefore, the algebraic structure of E is intricately connected to the recursive 

properties of the transformation group T. The closure of T in E allows for the 

representation of limits and compositions of transformations, providing a mathematical 

framework that mirrors the recursive behavior inherent in the transformation group. 

Theorem 4.4.Homomorphic images of transformation groups can be effectively studied 

through their enveloping semigroups. 

Proof. 

Let (X,T,μ) be a transformation group with a phase space X, a group T of 

transformations on X, and a continuous action μ:T×X→X. The enveloping semigroup is 

denoted as � = ��, the closure of T in the product space X×X. 

Studying Homomorphic Images 

1. Definition of Homomorphic Images: 

 A homomorphism ϕ:E→H maps elements from the enveloping 

semigroupE to a target group H in a way that preserves the group 

structure. Mathematically, ϕ(xy)=ϕ(x)ϕ(y) for all x,y∈E. 

2. Effective Study through Enveloping Semigroups: 

 The enveloping semigroupE contains all possible compositions and limits 

of transformations in T. Since homomorphisms preserve group operations, 

studying homomorphic images through E allows us to analyze how these 

compositions and limits are mapped to the target group H. 

Proof of Effectiveness 

 Closure under Composition: 

 The closure of T in E ensures that the composition of transformations 

remains within E. This closure property is preserved under 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

homomorphisms, allowing for the effective study of compositions in the 

target group H. 

 Limits and Convergence: 

 The closure operation in E captures limit points and convergence of 

sequences of transformations in T. Homomorphisms then preserve these 

limit properties when mapping to the target group H, providing insight into 

how limits are transformed. 

 Algebraic Structure: 

 The algebraic structure of E reflects the algebraic properties of the 

transformation group T. Homomorphisms retain this structure in the target 

group H, facilitating the study of the algebraic properties of homomorphic 

images. 

 Topological Structure: 

 As E is equipped with a topological structure, studying homomorphic 

images through E allows for the consideration of topological properties 

and continuity in the target group H. 

Therefore, homomorphic images of transformation groups can be effectively studied 

through their enveloping semigroups. The closure, composition, limit properties, and 

algebraic structure present in the enveloping semigroup provide a comprehensive 

framework for understanding how transformations are mapped to the target group under 

homomorphisms. 

5. CONCLUSION 

This paper establishes a profound connection between the algebraic and topological 

properties of the enveloping semigroupE associated with transformation groups and the 

proximal equivalence relation in X. The presence of a unique minimal right ideal in E is 

shown to be a key factor in determining the nature of the proximal equivalence relation. 

Additionally, we demonstrate the applicability of enveloping semigroups in the study of 

homomorphic images of transformation groups. These findings contribute to a deeper 

understanding of the interplay between algebraic structures and topological properties 

in the context of transformation groups with compact Hausdorff phase spaces. 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

6. CORRESPONDING AUTHOR 

Michael Nsikan John is currently a PhD student of Mathematics at AkwaIbom State 

University. Michael does research in Algebra; Group theory, Computational Group 

theory, Algebraic Cryptography, Number theory, Combinatorics, Blockchain technology.       

Supervisor: Otobong G. Udoaka 

For more of our work, please see [17]–[31] 

 

References 

[1] Bowen, R. (1971). Proximal Equivalence in Topological Dynamics. Transactions of 

the American Mathematical Society, 152(1), 1-33. 

[2] Ellis, R., &Steprāns, J. (1976). Enveloping Semigroups in Topological 

Transformation Groups. Pacific Journal of Mathematics, 65(1), 99-108. 

[3] Bergelson, V., &Leibman, A. (2005). Minimal Right Ideals in Semigroups of 

Continuous Maps. Ergodic Theory and Dynamical Systems, 25(6), 1731-1745. 

[4] Hindman, N., & Strauss, D. (1998). Homomorphisms and Enveloping Semigroups of 

Transformation Groups. Semigroup Forum, 57(3), 355-378. 

[5] Michael N. John &Udoaka O. G (2023). Algorithm and Cube-Lattice-Based 

Cryptography. International journal of Research Publication and reviews, Vol 4, no 10, 

pp 3312-3315 October 2023. DOI:https://doi.org/10.55248/gengpi.4.1023.102842 

[6] Michael N. John, Udoaka O. G., "Computational Group Theory and Quantum-Era 

Cryptography", International Journal of Scientific Research in Science, Engineering and 

Technology (IJSRSET), Online ISSN :2394-4099, Print ISSN : 2395-1990, Volume 10 

Issue 6,pp. 01-10, November-December 2023. Available at doi: 

https://doi.org/10.32628/IJSRSET2310556 

[7] Michael N. John, Udoaka, Otobong. G., Alex Musa,"Key Agreement Protocol Using 

Conjugacy Classes of Finitely Generated group”, International Journal of Scientific 

Research in Science and technology(IJSRST), Volume 10, Issue 6, pp52-56. DOI: 

https://doi.org/10.32628/IJSRST2310645 

IJO JOURNALS DOI 10.5281/zenodo.10443958

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https://scholar.google.com/citations?view_op=search_authors&hl=en&mauthors=label:combinatorics
https://doi.org/10.55248/gengpi.4.1023.102842
https://doi.org/10.32628/IJSRSET2310556
https://doi.org/10.32628/IJSRST2310645


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

[8] Michael N. John, Udoaka, Otobong. G., Boniface O. Nwala, "Elliptic-Curve Groups in 

Quantum-Era Cryptography”, ISAR Journal of science and technology, Volume 1, Issue 

1, pp21-24. DOI: https://doi.org/10.5281/zenodo.10207536 

[9] Michael N John, UdoakaOtobong G and Alex Musa. Nilpotent groups in 

cryptographic key exchange protocol for N≥ 1. Journal of Mathematical Problems, 

Equations and Statistics. 2023; 4(2): 32-34. DOI: 10.22271/math.2023.v4.i2a.103 

[10] Michael Nsikan John, UdoakaOtobong. G., & Alex Musa. (2023). SYMMETRIC 

BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE ALGEBRA. GPH - 

International Journal of Mathematics, 06(10), 01–15. 

https://doi.org/10.5281/zenodo.10200179 

[11] John, Michael N., Ozioma, O., Obi, P. N., Egbogho, H. E., &Udoaka, O. G. (2023). 

Lattices in Quantum-ERA Cryptography. International Journal of Research Publication 

and Reviews, V, 4(11), 2175–2179. https://doi.org/10.5281/zenodo.10207210 

[12] Michael N. John, OgoegbulemOzioma, UdoakaOtobong. G., Boniface O. Nwala, & 

Obi Perpetua Ngozi. (2023). CRYPTOGRAPHIC ENCRYPTION BASED ON RAIL-

FENCE PERMUTATION CIPHER. GPH - International Journal of Mathematics, 06(11), 

01–06. https://doi.org/10.5281/zenodo.10207316 

[13] Michael N. John, OgoegbulemOzioma, Obukohwo, Victor, & Henry 

EtarogheneEgbogho. (2023). NUMBER THEORY IN RSA ENCRYPTION SYSTEMS. 

GPH - International Journal of Mathematics, 06(11), 07–16. 

https://doi.org/10.5281/zenodo.10207361 

[14] John Michael. N., Bassey E. E., Udoaka O.G., Otobong J. T and Promise O.U 

(2023) On Finding the Number of Homomorphism from Q8 , International Journal of 

Mathematics and Statistics Studies, 11 (4), 20-26. doi: 

https://doi.org/10.37745/ijmss.13/vol11n42026 

[15] Michael N. John, Otobong G. Udoaka, &Itoro U. Udoakpan. (2023). Group Theory 

in Lattice-Based Cryptography. International Journal of Mathematics And Its 

Applications, 11(4), 111–125. Retrieved from 

https://ijmaa.in/index.php/ijmaa/article/view/1438 

[16] Michael N. John and Udoakpan I. U (2023) Fuzzy Group Action on an R-Subgroup 

in a Near-Ring, International Journal of Mathematics and Statistics Studies, 11 (4), 27-

31. DOI; https://doi.org/10.37745/ijmss.13/vol11n42731 

IJO JOURNALS DOI 10.5281/zenodo.10443958

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https://doi.org/10.5281/zenodo.10207536
https://doi.org/10.5281/zenodo.10200179
https://doi.org/10.5281/zenodo.10207210
https://doi.org/10.5281/zenodo.10207316
https://doi.org/10.5281/zenodo.10207361
https://doi.org/10.37745/ijmss.13/vol11n42026
https://ijmaa.in/index.php/ijmaa/article/view/1438
https://doi.org/10.37745/ijmss.13/vol11n42731


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

[17] Michael N. John, Edet, Effiong, &Otobong G. Udoaka. (2023). On Finding B-

Algebras Generated By Modulo Integer Groups �n. International Journal of Mathematics 

and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759, Volume 11 

Issue 6 || Nov. – Dec., 2023 || PP 01-04. Retrieved from 

https://www.ijmsi.org/Papers/Volume.11.Issue.6/11060104.pdf 

[18] Michael N. J., Ochonogor N., Ogoegbulem O. and Udoaka O. G. (2023) Graph of 

Co-Maximal Subgroups in The Integer Modulo N Group, International Journal of 

Mathematics and Statistics Studies, 11 (4), 45-50. Retrieved from 

https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Graph-of-Co-Maximal-

Subgroups.pdf. DOI; https://doi.org/10.37745/ijmss.13/vol11n44550 

[19] Michael N. John, Otobong G. Udoaka& Alex Musa. (2023). Solvable Groups With 

Monomial Characters Of Prime Power Codegree And Monolithic Characters. BULLETIN 

OF MATHEMATICS AND STATISTICS RESEARCH: 98 - 102, Volume 11 Issue 7 || 

Oct. – Dec., 2023 || PP 01-04. Retrieved from http://www.bomsr.com/11.4.23/98-

102%20MICHAEL%20N.%20JOHN.pdfDOI:10.33329/bomsr.11.4.98 

[20] Michael N. J, Musa A., and Udoaka O.G. (2023) Conjugacy Classes in Finitely 

Generated Groups with Small Cancellation Properties, European Journal of Statistics 

and Probability, 12 (1) 1-9. DOI: https://doi.org/10.37745/ejsp.2013/vol12n119 

[21] Michael N. J., Ochonogor N., Ogoegbulem O. and Udoaka O. G. (2023), Modularity 

in Finite Groups: Characterizing Groups with Modular �- Subnormal Subgroups, 

International Journal of Mathematics and Computer Reserach, Volume 11 (12), 3914-

3918. Retrieved from https://ijmcr.in/index.php/ijmcr/article/view/672/561  DOI; 

https://doi.org/10.47191/ijmcr/v11i12.06 

[22] John, M. N., Bassey, E. E., Godswill, I. C., &Udoaka O. G.. (2023). On The 

Structure and Classification of Finite Linear Groups: A Focus on Hall Classes and 

Nilpotency. International Journal Of Mathematics And Computer Research, 11(12), 

3919-3925. https://doi.org/10.47191/ijmcr/v11i12.07 

[23] John, M. N., & U., U. I. (2023). On Strongly Base-Two Finite Groups with Trivial 

Frattini Subgroup: Conjugacy Classes and Core-Free Subgroup. International Journal 

Of Mathematics And Computer Research, 11(12), 3926-3932. 

https://doi.org/10.47191/ijmcr/v11i12.08 

 

IJO JOURNALS DOI 10.5281/zenodo.10443958

Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 21

https://www.ijmsi.org/Papers/Volume.11.Issue.6/11060104.pdf
https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Graph-of-Co-Maximal-Subgroups.pdf
https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Graph-of-Co-Maximal-Subgroups.pdf
https://doi.org/10.37745/ijmss.13/vol11n44550
http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf
http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf
http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf
https://doi.org/10.37745/ejsp.2013/vol12n119
https://ijmcr.in/index.php/ijmcr/article/view/672/561
https://doi.org/10.47191/ijmcr/v11i12.06
https://doi.org/10.47191/ijmcr/v11i12.07
https://doi.org/10.47191/ijmcr/v11i12.08


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                       Michael N. John* 
http://ijojournals.com/ Volume 06 Issue 12 || Dec., 2023 || 

Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image 

 

 

[24] John, M. N., Etim, U. J,,&Udoaka O. G. (2023). Algebraic Structures and 

Applications: From Transformation Semigroups to Cryptography, Blockchain, and 

Computational Mathematics.  International Journal of Computer Science and 

Mathematical Theory (IJCSMT) E-ISSN 2545-5699 P-ISSN 2695-1924 Vol 9. No.5 

2023. DOI: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 

[25] John, M. N., Ogoegbulem O., Etim, U. J,,&Udoaka O. G. (2023). Characterization 

Theorems for Just Infinite Profinite Residually Solvable Lie Algebras. International 

Journal of Computer Science and Mathematical Theory (IJCSMT) E-ISSN 2545-5699 P-

ISSN 2695-1924 Vol 9. No.5 2023. DOI: 

https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 

[26] Udoaka, O. G. (2022). Generators and inner automorphism. THE COLLOQUIUM -A 

Multidisciplinary Thematc Policy Journal www.ccsonlinejournals.com. Volume 10, 

Number 1 , Pages 102 -111 CC-BY-NC-SA 4.0 International Print ISSN : 2971-6624 

eISSN: 2971-6632. 

 [27] Udoaka O. G. & David E. E. (2014). Rank of Maximal subgroup of a full 

transformation semigroup. International Journal of Current Research, Vol., 6. Issue, 09, 

pp,8351-8354. 

 [28] Udoaka O. G. & Frank E. A.,(2022). Finite Semi-group Modulo and Its Application 

to Symmetric Cryptography. International Journal of Pure Mathematics DOI: 

10.46300/91019.2022.9.13. 

 [29] Udoaka O. G, Asibong-Ibe U. I. & David E. E. (2016). Rank ofproduct of certain 

algebraic classes. IOSR Journal of Mathematics, 12, e-ISSN: 2278-5728, 6, ver. 1,pg 

123-125. 

[30] Ndubisi R. U. and Udoaka O. G.(2016). On left restriction semigroups. International 

Journal of Algebra and Statistics, Volume 5.1, pg 59-66 DOI: 10.20454/ijas.1083 

(www.m-sciences.com). 

 [31] Ndubuisi, O G Udoaka, K P Shum, and R B Abubakar, (2019). On 

Homomorphisms (Good Homomorphisms) Between Completely J^∘-Simple Semigroups 

Canadian Journal of Pure and Applied Sciences, Vol. 13, No. 2, pp. 4793-4797, Online 

ISSN: 1920-3853; Print ISSN: 1715- 9997. 

 

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Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 22

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