WORK PERFORMED BY CLOSED AND CLOPENM-TOPOLOGICAL FULL TRANSFORMATION SEMIGROUPSPACES MCTn AND Clp(MCTn)

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 MTn , the closed m-topological semigroup MCTn , and its clopen counterpart Clp(MCTn ), 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 bxc = {n | n ∈ Z +, n ≤ x < n+ 1}. Numerical evaluations confirm the validity of the derived formulas and reveal consistent growth patterns, thereby highlighting new combinatorial properties of m-topological transformation semigroup spaces.