(untitled) (Analysis in Theory and Applications, 2022)
For a finite discrete topological space X with at least two elements, a nonempty set Γ, and a map φ:Γ → Γ, σ_φ:X^Γ → X^Γ with σ_φ((x_α)_α Γ)=(x_φ(α))_α Γ (for (x_α)_α Γ X^Γ) is a generalized shift. In this text for S = {σ_φ:φ Γ^Γ} and H={σ_φ:Γ φ Γ is bijective} we study proximal relations of transformation semigroups (S, X^Γ) and (H, X^Γ). Regarding proximal relation we prove: P(S, X^Γ) = {((x_α)_α Γ,(y_α)_α Γ) X^Γ × X^Γ : β Γ (x_β = y_β)}and P(H, X^Γ ) {((x_α)_α Γ,(y_α)_α Γ) X^Γ × X^Γ : {β Γ : x_β = y_β} is infinite} {( x,x) : x X}. Moreover, for infinite Γ, both transformation semigroups (S, X^Γ) and (H, X^Γ) are regionally proximal, i.e., Q(S, X^Γ) = Q(H, X^Γ ) = X^Γ × X^Γ, also for sydetically proximal relation we have L(H, X^Γ) = {((x_α)_α Γ,(y_α)_α Γ) X^Γ × X^Γ : {γ ∈ Γ : x_γ eq y_γ} is finite}.