A Note on Transformation Semigroups
M Sabaghan, Fatemah Ayatollah Zadeh Shirazi · 2001
In this note we study the transformation semigroup (X,S), where S is a finite union of its subsemigroups. 2000 AMS Classification Subject: 54H15 By a transformation semigroup (X,S,?X ) (or simply (X,S)) we mean a compact Hausdorff topological space X, a discrete topological semigroup S with identity e and a continuous map ?σ: X×S→ X (?σ(x, s) = xs (∀ x∈ X, ∀ s∈ S)) such that: •∀ x∈ X xe =x, •∀ x∈ X ∀ s, t∈ S x (st) = (xs) t. In the transformation semigroup (X,S) we have the following definitions: 1. For each s∈ S, define the continuous map ?⌡ s : X→ X by x?⌡ s =xs (∀ x∈ X), we used to write s instead of ? s . The closure of {? s │s∈S} in X X with pointwise convergence, is called the enveloping sermigroup (or Ellis semigroup) of (X,S) and it is written by E(X,S) or simply E(X). E(X,S) has a semigroup structure (Ellis, 1969, Chapter 3), a nonempty subset K of E(X,S) is called a right ideal if KE(X,S) ⊆ K, and it is called a minimal right ideal if none of the right ideals of E(X,S) is a proper subset of K. 2. A nonempty subset Z of X is called invariant if ZS ⊆ Z, moreover it is called minimal if it is closed and none of the closed invariant subsets