On Finite Structures of Cyclicpoid Classes and Composed Functions in Star-Like Transformation Semigroups
DOI:
https://doi.org/10.53560/PPASA(63-3)716Keywords:
Monogenic Semigroup, Cayley Table, Star-like Cyclicpoid, Transformation Semigroup, Tropical Algebra, Semigroup KernelAbstract
This research explores the algebraic and structural characteristics of finite star-like transformation semigroups that have cyclic features. We formalize the concept of cyclicpoid classes, characterized as star-like partial transformations that function alongside single-valued unary inversion operations. Concerning the structural deviation from classical cyclic groups and monogenic semigroups, specifically, we examine the systems where their kernels do not constitute group subsemigroups. By employing generator-index-period parameters joint capacity parameters , and subsemigroup kernel decompositions, we establish a lower-bound theorems that regulate operational transformation paths (Theorem 3). Additionally, by representing composition matrices in real min-plus algebras, we create a tropical polynomial form to ascertain exact tropical roots (eigenvalues). A non-group Cayley table with 4 elements, an evaluation of a tropical matrix eigenvalues, and applications of combinatorial parameters are provided to validate the theoretical framework. This study integrates star-like partial function limits with tropical spectral descriptions for generalized cyclic transformation semigroups.
References
D. Maclagan and B. Sturmfels (Eds.). Graduate Studies in Mathematics: Introduction to tropical geometry. American Mathematical Society, Providence, Rhode Island, USA (2015). https://doi.org/10.1090/gsm/161
I. Itenberg, G. Mikhalkin and E. Shustin (Eds.). Tropical Algebraic Geometry (Second Edition). Birkhäuser Basel, Switzerland (2009). https://doi.org/10.1007/978-3-0346-0048-4
P.A. Helminck and Y. Ren. Generic root counts and flatness in tropical geometry. Journal of the London Mathematical Society 111(5): e70171 (2025). https://doi.org/10.1112/jlms.70171
A.J. Puentes, H. Markwig, S. Pauli, and F. Röhrle. Quadratically enriched plane curve counting via tropical geometry. arXiv preprint arXiv: 2502.02569 (2025). https://doi.org/10.48550/arXiv.2502.02569
G.R. Ibrahim, G.N. Bakare, I.F. Usamot, and T.O. Aliu. Tropical geometric approach to signed full transformation semigroups. The Transactions of the Nigerian Association of Mathematical Physics 21: 205-212 (2025). https://doi.org/10.60787/tnamp.v21.512
E. Katz. Lifting tropical curves in space and linear systems on graphs. Advances in Mathematics 230(3): 853-875 (2012). https://doi.org/10.1016/j.aim.2012.03.017
E. Brugallé and K. Shaw. A bit of tropical geometry. The American Mathematical Monthly 121(7): 563-589 (2014). https://doi.org/10.4169/amer.math.monthly.121.07.563
K. Shaw. A tropical intersection product in matroidal fans. SIAM Journal on Discrete Mathematics 27(1): 459491 (2013). https://doi.org/10.1137/110850141
I.F. Usamot, G.N. Bakare, and B.M. Ahmed. Tropicalization of idempotent elements of full transformation semigroups. Confluence Journal of Pure and Applied Sciences 2(1): 56-61 (2018).
http://cjpas.org.ng/index.php/pub/article/view/33
U.F. Idayat, I.G. Risqot, B.G. Naimat, A.A. Dauda, and T.O. Aliu. Green’s equivalence classes in full transformation semigroups: A height-based approach. Euler: Jurnal Ilmiah Matematika, Sains dan Teknologi 13(1): 45-49 (2025). https://doi.org/10.37905/euler.v13i1.30957
G.R. Ibrahim, G.N. Bakare, I.F. Usamot, O.I. Sulyman, S.A. Akinwunmi, and A.D. Adeshola. Tropical roots and their multiplicities on the subsemigroup of order-decreasing and order-preserving full transformation. Lasu Journal of Research and Review in Science 11: 121-133 (2024). https://kwasuspace.kwasu.edu.ng/handle/123456789/6113
B. Ali, M.A. Jada, and M.M. Zubairu. On the ranks of certain semigroups of order preserving partial isometries of a finite chain. Journal of Algebra and Related Topics 6(2): 15-33 (2018). http://doi.org/10.22124/jart.2019.10875.1109
G.U. Garba. On the idempotent ranks of certain semigroups of order-preserving transformation. Glasgow Mathematical Journal 36(1): 1-9 (1994). http://doi.org/10.1017/S0017089500030482
J.M. Howie. Fundamentals of semigroup theory. Semigroup Forum 58: 156-158 (1999). https://doi.org/10.1007/s002339900006
O. Ganyushkin and V. Mazorchuk (Eds.). Classical finite transformation semigroups: An introduction. Springer-Verlag, London (2009). https://doi.org/10.1007/978-1-84800-281-4
P.M. Higgins. Combinatorial results for semigroups of order-preserving mappings. Mathematical Proceedings of the Cambridge Philosophical Society 113(2): 281-296 (1993). https://doi.org/10.1017/s0305004100075964
S.A. Akinwunmi. The cyclicpoid life stages of a star-like mathematician: The cyclicpoid life stages. Journal of the Nigerian Mathematical Society 44(1): 139-151 (2025). http://ojs.ictp.it/jnms/index.php/jnms/article/view/1126
S.A. Akinwunmi, M.M. Mogbonju, and A.O. Adeniji. Multiplicative invertibility characterizations on star-like cyclicpoid C_y Pω_n^* finite partial transformation semigroups. Hikari Journal of Pure Mathematical Science 10(1): 45-55 (2021). https://doi.org/10.12988/pms.2021.91276
S.A. Akinwunmi, G.R. Ibrahim, A.O. Adeniji, and D.O. Oyewola. On the multiplicity order of spinnable star-like transformation semigroup. Journal of the Nigerian Society of Physical Sciences 6(1): 1692 (2024). https://doi.org/10.46481/jnsps.2024.1692
A.T. Imam and M.J. Ibrahim. On products of 3-paths in finite full transformation semigroups. Algebra and Discrete Mathematics 33(2): 60-77 (2022). https://doi.org/10.12958/adm1770
A. Umar. Some combinatorial results in the theory of partial transformation semigroups. Algebra and Discrete Mathematics 17(1): 110-134 (2014). http://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027/551
A. Kachalla and B.A. Madu. Expressing partial order-preserving transformations as products of nilpotent. International Journal of Development Mathematics 1(3): 112-125 (2024). https://doi.org/10.62054/ijdm/0103.08
P. Srimora, R. Chinram and W. Yonthanthum. Regularity of the transformation semigroups with invariant sets and restricted by an equivalence relation. Gulf Journal of Mathematics 20(2): 194-202 (2025). https://doi.org/10.56947/gjom.v20i2.3191
R. Chinram, P. Petchkaew, S. Baupradist. Left and right magnifying elements in generalized semigroups of transformations by using partitions of a set. European Journal of Pure and Applied Mathematics 11(3): 580-588 (2018). https://doi.org/10.29020/nybg.ejpam.v11i3.3260
A. H. Clifford and G. B. Preston (Eds.). Mathematical Surveys: The Algebraic Theory of Semigroups. American Mathematical Society, Providence, Rhode Island, USA (1961). https://doi.org/10.1090/surv/007.1
S. Lipscomb (Ed.). Mathematical Surveys and Monographs: Symmetric Inverse Semigroups. American Mathematical Society, Providence, Rhode Island, USA (1996). https://doi.org/10.1090/surv/046
