AN ENVELOPING SEMIGROUP PROOF OF THE FACT THAT RP [d] IS AN EQUIVALENCE RELATION

Let T be a countable abelian group and let (X,T ) be a minimal dynamical system; i.e. X is a compact Hausdorff space and T acts on it as a group of homeomorphisms. Following [4] and [7] we introduce the following notations (generalizing from the case T = Z to the case of a general T action). For an integer d ≥ 1 let X [d] = X2d . We index the coordinates of an element x ∈ X [d] by subsets ⊂ {1, . . . , d}. Thus x = (x : ⊂ {1, . . . , d}), where for each , x ∈ X = X. E.g. for d = 2 we have x = (x∅, x{1}, x{2}, x{1,2}). We let X [d] ∗ = X d−1 = ∏ {X : 6= ∅} and for x ∈ X [d] we let x∗ ∈ X [d] ∗ denote its projection; i.e. x∗ is obtained by omitting the ∅-coordinate of x. For each ⊂ {1, . . . , d} we denote by π the projection map from X [d] onto X = X. For a point x ∈ X we let x ∈ X [d] and x ∗ ∈ X [d] ∗ be the diagonal points all of whose coordinates are x. ∆ = {x : x ∈ X} is the diagonal of X [d] and