Green’s Equivalence Classes in Full Transformation Semigroups: A Height-based Approach

Usamot Foluke Idayat; Ibrahim Garba Risqot; Bakare Gatta Naimat; Adeshola Adediran Dauda; Aliu Tajudeen Olaniyi

doi:10.37905/euler.v13i1.30957

Green’s Equivalence Classes in Full Transformation Semigroups: A Height-based Approach

Usamot Foluke Idayat, Ibrahim Garba Risqot, Bakare Gatta Naimat, Adeshola Adediran Dauda, Aliu Tajudeen Olaniyi

Abstract

Green’s relations constitute five distinct equivalence classes that serve to define the elements of a semigroup. In this work, the equivalence classes within the Full Transformation Semigroup (Tκ) were systematically enumerated according to their respective heights, and the findings derived were subsequently generalized through the application of a combinatorial assertion and their graphical representations included to substantiate the theoretical results.

Keywords

Semigroup; Full transformation; Green’s relations; Equivalence classes; Combinatorial

Full Text:

PDF

References

J. A. Green, "On the structure of semigroups," Annals of Mathematics, vol. 54, pp. 163–172, 1951. doi: 10.2307/1969312.

J. M. Howie, "Semigroups, past, present and future," in Proc. of the International Conference of Algebra and its Applications, 2002, pp. 6–20.

A. Umar, "On the semigroup of partial one-to-one order-decreasing finite transformations," Proc. Soc., vol. 123A, pp. 355–363, 1993. doi: 10.1017/S0308210500025737.

I. F. Usamot, G. N. Bakare, and B. M. Ahmed, "Tropicalization of idempotent elements on full transformation semigroup," Confluence Journal of Pure and Applied Sciences, vol. 2, no. 1, pp. 56–61, 2018.

C. Miller, "The R-height of semigroups and their bi-ideals," International Journal of Algebra and Computation, vol. 33, no. 1, pp. 47–66, 2022.

G. R. Ibrahim, G. N. Bakare, S. A. Akinwunmi, and O. T. Aliu, "Application of lattice theory on order-preserving full transformation semigroup via fixed points," Ilorin Journal of Science, vol. 10, no. 2, pp. 82–86, 2023.

O. Meiqing and F. Xuerong, "Natural partial order on generalized semigroups of transformation," Wiley Journal of Mathematics, pp. 1–9, 2025.

J. East and P. Higgins, "Green’s relations and stability for subsemigroups," Semigroup Forum, vol. 101, pp. 77–86, 2020. doi: 10.1007/s00233-019-10039-8.

J. East, "Structure of principal one-sided ideals," International Journal of Algebra and Computation, vol. 31, pp. 1093–1145, 2021.

W. Rattiya and C. Yanisa, "Green’s relations and regularity on semigroups of partial transformations," Communications in Algebra, vol. 50, no. 9, pp. 3827–3839, 2022.

J. Awolola and M. Ibrahim, "Connections of Green’s relations of a Γ-semigroup with operator semigroups," Annals of Mathematics and Computer Science, vol. 22, 2024, pp. 81–87.

S. Janson and V. Mazorchuk, "Some remarks on the combinatorics of isn," Semigroup Forum, vol. 109, pp. 391–405, 2005.

G. N. Bakare, I. F. Usamot, B. M. Ahmed, and Y. T. Oyebo, "Combinatorial properties on subsemigroup of order-decreasing alternating semigroup using three distinct combinatorial functions; new approach to generalization," Daffodil International University Journal of Science and Technology, vol. 17, no. 3, pp. 29–35, 2022. doi: 10.5281/zenodo.13827986.

J. M. Howie, Introduction to the Fundamentals of Semigroup Theory, USA: Oxford University Press, 1995.

W. D. Munn, "The characters of the symmetric inverse semigroup," Proc. Cambridge Philos., vol. 53, pp. 13–18, 1957. doi: 10.1017/S0305004100031947.

G. U. Garba, "Nilpotents in semigroups of partial one-to-one order-preserving mappings," Semigroup Forum, vol. 48, pp. 37–49, 1994.

J. M. Howie and R. B. Mcfadden, "Idempotent rank in finite full transformation semigroups," Proc. of the Royal Society of Edinburgh Section A, vol. 114, pp. 161–167, 2005.

J. M. Howie, "Products of idempotents in certain semigroups of transformations," Proc. Edinb. Math. Soc., vol. 17, no. 3, pp. 223–236, 1971.

O. Ganyushkin and V. Mazorchuk, "On the structure of ion," Semigroup Forum, vol. 66, pp. 455–483, 2003.

A. Laradji and A. Umar, "On the number of nilpotents in the partial symmetric semigroup," Communications in Algebra, vol. 32, pp. 3017–3023, 2004.

N. J. A. Sloane, "The online encyclopedia of integer sequences." [Online]. Available: http://www.research.att.com/njas/sequences/.

J. M. Howie, "The subsemigroup generated by the idempotents of a full transformation semigroup," J. London Math. Soc., vol. 41, pp. 707–716, 1966.

J. M. Howie, "Semigroups of Mappings," Technical Report Series, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia, 2006, pp. 1–32.

DOI: https://doi.org/10.37905/euler.v13i1.30957