Abstract
Let Tn be the set of full transformations and Pn be the set of partial transformations. It is shown that Tn form a semi-group of order nn and Pn form a semi-group of order(n+1)n. Let ρ(n,m) be a binary relation then we define the image set of ρ(n,m), I(ρ): {n|n∈Nandthereexistsm∈M:(m,n)∈ρ} whenever (≡ρ): ≡ρ on a set M is called an equivalence relation if ≡ρ is reflexive, symmetric and transitive. Then, For all m∈M, we let [m] equivalence class denote the set [m]={n∈M|n≡ρm} with respect to ≡ρ determine by m. Furthermore, we show that D=L∘R=R∘L=LυR implies L⊆J and R⊆J. Therefore, D is the minimum equivalence relation class containing L and R. Hence, D⊆J. If n∈Xm:{n∈Xm|nxn=n;nx=xn} then n∈Dclass is regular. We also show that for Lclass, Rclass and Hclass for all m,n∈D(α) we have α such that D(α)⊆M implies I(α)⊆M. Then for any transformation of a finite semi-group β,α∈S where α=(123123) and β=(123321), α and β represent the five equivalence relations and CPn represent the sub-semi-group of partial contraction mapping onM={1,2,3…} while |Q| denotes the order ofQ.
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