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 IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 9 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 IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 10 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 IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 11 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: IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 12 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→∞. IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 13 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. IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 14 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. IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 15 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. IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 16 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 IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 17 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. IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 18 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 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 19 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 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 20 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. IJO JOURNALS DOI 10.5281/zenodo.10443958 Volume 06 | Issue 12 | December 2023 | http://ijojournals.com/index.php/m/index 22 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 http://www.m-sciences.com/