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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.

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