Year: 2025
Type: article
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.
Authors: M. A. Mbah, Eze C., Pokalas P. Tal, S. Kasim
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4411124661