AVERAGE AND EXTREMAL VALUES OF COMPLETE WORK-DONE ON TRANSFORMATION SEMIGROUPS

Abstract In this paper, we study a complete work-done function defined on three important transformation semigroups: the full transformation semigroup (Tn), the partial transformation semigroup (Pn), and the symmetric inverse semigroup (In). For each class, closed formulae are obtained for the total and average values of the complete work-done function. Extremal values are also determined for (Tn) and (Pn). The results show that algebraic restrictions such as partiality and injectivity significantly influence the magnitude and distribution of the complete work-done values. Numerical tables and graphical illustrations are provided for1 ≤ n ≤ 10 to support the theoretical findings. MSC (2020): Primary 05A15, Secondary 20M10, 20M20