WORK PERFORMED BY CLOSED AND CLOPEN M-TOPOLOGICAL FULL TRANSFORMATION SEMIGROUP SPACES MCTn AND Clp(MCTn )

MOSES OBINNA  FRANCIS; ADENIKE OLUSOLA  ADENIJI

MOSES OBINNA FRANCIS DEPARTMENT OF MATHEMATICAL SCIENCES, BINGHAM UNIVERSITY KARU, NASARAWA, NIGERIA.
ADENIKE OLUSOLA ADENIJI DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ABUJA, P.M.B 117, ABUJA-FCT, NIGE- RIA.

Keywords: Clopen, Closed, Transformation semigroup, work Performed

Abstract

Let α and β be two elements of an m-topological transformation semigroup space. In this paper, we introduce and derive explicit formulas for the work performed by elements in the full transformation semigroup M_Tn , the closed m-topological semigroup M_CTn , and its clopen counterpart Clp(M_CTn ), where the displacement of a point x under a transformation α is given by d(x, αx) = |x − αx|. We further establish explicit formulas for the average work and the power within these semigroup spaces. To ensure that the system stabilizes to integer values, we incorporate the floor function [x♩ = {n | n ∈ Z⁺, n ≤ x < n + 1}. Numerical evaluations confirm the validity of the derived formulas and reveal consistent growth patterns, thereby high- lighting new combinatorial properties of m-topological transformation semi- group spaces.

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