On Distality of a Transformation Semigroup with One Point Compactification of a Discrete Space as Phase Space

For infinite discrete topological space Y , suppose A ( Y ) is one point compactification of Y , in the following text we prove that the transformation semigroup ( A ( Y ) , S ) is distal if and only if the enveloping semigroup E ( A ( Y ) , S ) is a group of homeomorphisms on A ( Y ) , or equivalently for all p ∈ E ( A ( Y ) , S ) , p : A ( Y ) → A ( Y ) is pointwise periodic. Also, the transformation group ( A ( Y ) , S ) is distal (resp. equicontinuous, pointwise minimal) if and only if for all x ∈ A ( Y ) , x S is a finite subset of A ( Y ) . The text is motivated with tables, counterexamples and studying finally distality (and co-decomposability to distal transformation semigroups) in the abelian transformation semigroup ( A ( Y ) , S ) .