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All Journal Mikailalsys Journal of Mathematics and Statistics
Kasim, S.
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Engineering Mathematics Mechanical Engineering

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Graph-Theoretic Characterization of Quasi-Nilpotent Elements in Finite Semigroups of Full Order-Preserving Transformations C., Eze; O., Olaiya O.; Kasim, S.
Mikailalsys Journal of Mathematics and Statistics Vol 3 No 2 (2025): Mikailalsys Journal of Mathematics and Statistics
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjms.v3i2.5906

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.
A Graph-Theoretic Characterization of Orbits in the Finite Full Transformation Semigroup Mbah, M. A.; C., Eze; Tal, Pokalas P.; Kasim, S.
Mikailalsys Journal of Mathematics and Statistics Vol 3 No 3 (2025): Mikailalsys Journal of Mathematics and Statistics
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjms.v3i3.6073

Abstract

This paper investigates the orbit structures of elements in the full transformation semigroup TnT_n through the framework of digraph connectivity. Transformations are characterized based on whether their associated functional digraphs are strongly connected, weakly connected, or unilateral. It is shown that strong connectivity corresponds precisely to transformations whose orbits form a single nn-cycle. In contrast, unilateral connectivity arises when orbits constitute directed paths terminating in a unique cycle, and weak connectivity is identified when all elements belong to a single weakly connected component. Furthermore, the paper provides enumeration results, proving that there are exactly (n−1)!(n - 1)! transformations with strongly connected (cyclic) orbits and n!(n−1)n!(n - 1) transformations with unilateral orbit structures. These findings offer new structural and enumerative insights into the full transformation semigroup by analyzing the connectivity patterns of its orbit representations.
Co-Authors C., Eze Mbah, M. A. O., Olaiya O. Tal, Pokalas P.