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All Journal Mikailalsys Journal of Mathematics and Statistics
O., Olaiya O.
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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.
On the Connectivity and Eulerian Properties of Cayley Digraphs in Transformation Monoids Danlami, Kazaik Benjamin; Eze, Chibueze; O., Olaiya O.
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.v3i3.6134

Abstract

This paper establishes fundamental connections between the graph-theoretic properties of Cayley digraphs and the algebraic structure of transformation monoids. Our main contributions include a complete characterization of strong connectivity in transformation monoids, proving that for a transformation monoid T acting on a finite set X, the Cayley digraph Cay(T, S) with respect to a generating set S ⊆ T is strongly connected if and only if T contains the full symmetric group S(X); and a classification of Eulerian properties in symmetric groups, demonstrating that for the symmetric group Sn with any generating set S, the Cayley digraph Cay(Sn, S) is Eulerian precisely when it is strongly connected. We provide concrete examples illustrating these theorems, including detailed Cayley graph constructions for S3 with explicit generating sets. Our results reveal deep connections between monoid theory and graph theory, showing how algebraic properties manifest in combinatorial structures. The proofs employ techniques from semigroup theory, algebraic graph theory, and finite group theory, offering new insights into the representation of transformation monoids through their generator-dependent digraphs. This work contributes to the broader understanding of how algebraic structures can be studied through their associated graphs.
Co-Authors C., Eze Danlami, Kazaik Benjamin Eze, Chibueze Kasim, S.