Graph-Theoretic Characterization of Quasi-Nilpotent Elements in Finite Semigroups of Full Order-Preserving Transformations
Article Sidebar
Page Numbers:
490-499
Digital Object Identifier:
10.58578/mjms.v3i2.5906
Viewed: 218 times
Downloaded: 148 times
LYAS Publisher cordially invites qualified professionals to serve as Editors or Reviewers. Your expertise will make an important contribution to maintaining and strengthening the academic quality of our publications. Interested applicants are kindly requested to complete the application form at the following link: Editors & Reviewers
Please feel free to contact us if you need any further information about the submission process or if you have any additional questions.
Authors:
Eze C.1*, Olaiya O. O.2, S. Kasim3 
Copyright :
Authors retain copyright and grant the journal right of first publication.
Main Article Content
Abstract
This paper investigates the structural behavior of quasi-nilpotent elements within the semigroup On of all full order-preserving transformations on a finite chain Xn = {1, 2, . . ., n}. While quasi-nilpotency has been extensively studied in full and partial transformation semigroups, its characterization in On remains largely unexplored. By employing a graph-theoretic approach, we associate to each transformation α ∈ On a digraph Γ(α) and establish necessary and sufficient conditions under which α is quasi-nilpotent. Specifically, we show that α is quasi-nilpotent if and only if Γ(α) has a unique sink and all vertices are connected to it via directed paths. This char- acterization is further refined by relating the height of Γ(α) to the number of convex blocks in the domain partition of α. Illustrative examples and explicit constructions are provided to validate the theoretical findings. The results offer new insights into the interplay between algebraic properties of transformation semigroups and their combi- natorial representations.
Downloads
Article Details
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
References
Howie, J.M. (2006). Semigroup of mappings. Technical report series, King Fahd University, Dhahran, Saudi Arabia.
Frank Harary (1959). The number of functional digraphs. Math. Ann, (1959).
Howie, J. M. (1995). Fundamentals of Semigroup Theory. Oxford University Press.
Clifford, A. H., & Preston, G. B. (1961). The Algebraic Theory of Semigroups. American Mathematical Society.
Ganyushkin, O., & Mazorchuk, V. (2008). Classical Finite Transformation Semi- groups: An Introduction. Springer.
Higgins, P. M. (1992). Techniques of Semigroup Theory. Oxford University Press.
Madu, B. A. (1999). Quasi-idempotents and quasi-nilpotents in finite transforma- tions semigroups (Unpublished doctoral dissertation). Ahmadu Bello University, Zaria-Nigeria.
Umar, A. (1993). On the semigroup of partial one-one order-decreasing finite trans- formation. Proceedings of the Royal Society of Edinburgh, 123A, 355–363.
Howie, J. M. (1971). Products of idempotents in certain semigroups of transfor- mations. Proceedings of the Royal Society of Edinburgh, 17A, 233–236.
Howie, J. M. (1966). The subsemigroup generated by the idempotents of a full transformation semigroup. Journal of the London Mathematical Society, 41, 707– 716.
Garba, G. U., Tanko, A. I., & Madu, B. A. (2011). Products and rank of quasi- idempotents in finite full transformation semigroups. JMI International Journal of Mathematical Sciences, 2(1), 12–19.