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All Journal Mikailalsys Journal of Mathematics and Statistics
Tal, Pokalas P.
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Engineering Mathematics Mechanical Engineering

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Maximum Works Performed by Signed Partial Transformations of a Finite Set Tal, Pokalas P.; Chibueze, Eze; Sanda, Yulari
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.v3i3.5962

Abstract

Let Xn and Xn* be the finite sets {1, 2, 3, ..., n} and {±1, ±2, ±3, ..., ±n} respectively. A map from Xn to Xn is called a transformation on Xn. We call a map a signed transformation if it maps from Xn to Xn*. Let Pn~ be the set of all signed partial transformations on Xn. This set consists of all transformations in Pn~ for which the domain of the transformation is a subset of Xn. The work w(alpha) performed by a transformation alpha is defined as the sum of all distances |i - alpha(i)| for each i in the domain of alpha. In this paper, we characterize all transformations in Pn~ that attain maximum and minimum works, and we deduce formulas for these minimum and maximum values. We further present a range for the values of w(alpha) for all transformations in Pn~.
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 Chibueze, Eze Kasim, S. Mbah, M. A. Sanda, Yulari