Dynamical equivalences on

For every infinite discrete group G, the remainder G ∗ = β G ∖ G of the Stone–Čech compactification βG of G has a natural structure of G-space. The orbit equivalence E on G ∗ ( ( x , y ) ∈ E ⇔ g x = y for some g ∈ G ) produces the following three derived equivalences on G ∗ ) y ) ∈ E ˚ ⇔ cl E x = cl E y , where E x , E y are E-equivalence classes containing x and y, ) s the smallest by inclusion equivalence on G ∗ containing E such that every E ˙ -equivalence class is closed, ) s the smallest by inclusion closed in G ∗ × G ∗ equivalence on G ∗ containing E, he relation ( E ˆ ) y ) ∈ E ˆ ⇔ ∃ z : x ∈ cl E z , y ∈ cl E z , is an equivalence if G is countable. dy the interrelations between the classes of these equivalences and the principal left ideals of the semigroup βG.