Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image
Abstract
This paper investigates the algebraic properties of the enveloping semigroup E 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.
References
Bowen, R. (1971). Proximal Equivalence in Topological Dynamics. Transactions of the American Mathematical Society, 152(1), 1-33.
Ellis, R., &Steprāns, J. (1976). Enveloping Semigroups in Topological Transformation Groups. Pacific Journal of Mathematics, 65(1), 99-108.
Bergelson, V., &Leibman, A. (2005). Minimal Right Ideals in Semigroups of Continuous Maps. Ergodic Theory and Dynamical Systems, 25(6), 1731-1745.
Hindman, N., & Strauss, D. (1998). Homomorphisms and Enveloping Semigroups of Transformation Groups. Semigroup Forum, 57(3), 355-378.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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.
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.
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.
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).
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.
Author(s) and co-author(s) jointly and severally represent and warrant that the Article is original with the author(s) and does not infringe any copyright or violate any other right of any third parties and that the Article has not been published elsewhere. Author(s) agree to the terms that the IJO Journal will have the full right to remove the published article on any misconduct found in the published article.
