On distality of a transformation semigroup with one point compactification of a discrete space as phase space

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Document Type : Regular Paper

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1 Faculty of Mathematics, Statistics and Compyter Sience, College of Science, University of Tehran, Tehran, Iran

2 Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran

Abstract

For infinite discrete topological space Y, suppose A(Y) is one point compactification of , 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), xS 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 abelian transformation semigroup (A(Y) , S).

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Iranian Journal of Science

Articles in Press, Accepted Manuscript
Available Online from 13 September 2015

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