© Hindawi Publishing Corp. a-MINIMAL SETS AND RELATED TOPICS IN TRANSFORMATION

Abstract. We give some generalizations of proximal relation and distal structure relation of a transformation semigroup in terms of A-minimal sets and A-minimal sets instead of minimal right ideals and conclude similar results. 2000 Mathematics Subject Classification. Primary 54H15. 1. Preliminaries. By a transformation semigroup (X,S,ρ) (or simply (X,S)) we mean a compact Hausdorff topological space X, a discrete topological semigroup S with identity e, and a continuous map ρ: X × S → X (ρ(x,s) = xs∀x ∈ X,∀s ∈ S), such that (1) xe = x ∀x ∈X; (2) x(st) = (xs)t ∀x ∈X, ∀s,t ∈ S. In the transformation semigroup (X,S), for each s ∈ S defineπs:X →X byπs(x)= xs (∀x ∈ X). We assume the semigroup S acts effectively on X, that is, for each s,t ∈ S, s ≠ t if and only if πs ≠πt. The closure of {πs | s ∈ S} in XX (with pointwise convergence topology) is called the enveloping semigroup (or Ellis semigroup) of (X,S) and is denoted by E(X,S) (or simply E(X)), E(X) has a semigroup structure [1]. A nonempty subset I of E(X) is called a right ideal of E(X) if IE(X) ⊆ I, moreover, if the right ideal I of E(X) does not have any proper subset which is a right ideal of E(X), then I is called a minimal right ideal of E(X), the set of all minimal right ideals of E(X) is denoted by Min(E(X)). An element u of E(X) is called idempotent if u2 = u