Some characterizations of equivalence relation on contraction mappings

Let T-n be the set of full transformations and P-n be the set of partial transformations. It is shown that T-n form a semi-group of order n(n) and P-n form a semi-group of order (n + 1)(n). Let rho (n, m) be a binary relation then we define the image set of rho (n, m), I(rho): {n vertical bar n is an element of N and there exists m is an element of M: (m, n) is an element of rho} whenever ((math) rho): (math) rho on a set M is called an equivalence relation if (math) rho is reflexive, symmetric and transitive. Then, For all m is an element of M, we let [m] equivalence class denote the set [m] = {n is an element of M vertical bar n (math) rho m} with respect to (math) rho determine by m. Furthermore, we show that D = L circle R = R circle L = L upsilon R. implies L subset of J and R subset of J. Therefore, D is the minimum equivalence relation class containing L and R. Hence, D subset of J. If n is an element of X-m: {n is an element of X-m vertical bar n x n = n; nx = xn} then n is an element of D-class is regular. We also show that for L-class, R-class and H-class for all m, n is an element of D(alpha) we have alpha such that D(alpha) subset of M implies I(alpha) subset of M. Then for any transformation of a finite semi-group beta, alpha is an element of S where alpha = [GRAPHICS] and beta = [GRAPHICS] , alpha and beta represent the five equivalence relations and CPn represent the sub-semi-group of partial contraction mapping on M = {1, 2, 3 . . .} while vertical bar Q vertical bar denotes the order of Q. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of African Institute of Mathematical Sciences / Next Einstein Initiative.