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

For infinite discrete topological space Y, Y, suppose A(Y) A(Y) is one point ompactification of Y, Y, in the following text we prove that the transformation semigroup (A(Y),S) (A(Y),S) is distal if and only if the enveloping semigroup E(A(Y),S) E(A(Y),S) is a group of homeomorphisms on A(Y), A(Y), or equivalently for all p∈E(A(Y),S) p∈E(A(Y),S), p:A(Y)→A(Y) p:A(Y)→A(Y) is pointwise periodic. Also, the transformation group (A(Y),S) (A(Y),S) is distal (resp. equicontinuous, pointwise minimal) if and only if for all x∈A(Y) x∈A(Y), xS xS is a finite subset of A(Y) 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) (A(Y),S).