Research Article

Height and Fixed Points in m-Topological Transformation Semigroup Spaces

Abstract

We investigate two enumerative invariants in three m-topological transformation semigroup spaces associated with full finite transformations, namely MTn , MCTn and Clp(MCTn ). The first invariant is the height distribution, which classifies elements by image size. Using a sequence model over an extended alphabet, we reduce the height problem to classical surjection counting and derive explicit closed formulas via inclusion–exclusion; in the clopen setting the enumeration simplifies to a Stirling-number expression. The second invariant is the fixed-point distribution, which counts transformations (and complements) according to the number of fixed points. For each space we obtain closed counting formulas, establish supporting identities and recurrences, and present numerical tables for 1 ≤ n ≤ 8. Illustrative plots for n = 8 show that height counts concentrate at intermediate image sizes, while fixed-point counts are largest for small numbers of fixed points, and they highlight how the closed and clopen restrictions shift these distributions relative to the full case.