Some characterizations of equivalence relation on contraction mappings

Abstract

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 ρ (n, m) be a binary relation then we define the image set of ρ (n, m), I (ρ):{n| n∈ N a n d t h e r e e x i s t s m∈ 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≡ ρ m} with respect to≡ ρ determine by m. Furthermore, we show that D= L∘ R= R∘ L= L υ R implies L⊆ J and R⊆ J. Therefore, D is the minimum equivalence relation class containing L and R. Hence, D⊆ J. If n∈ X m:{n∈ X m| n x n= n; n x= x n} then n∈ D c l a s s is regular. We also show that for L c l a s s, R c l a s s and H c l a s s for all m, n∈ D (α) we have α such that D (α)⊆ M implies I (α)⊆ M. Then for any transformation of …