Some characterizations of equivalence relation on contraction mappings

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.