On Proximal Relations in Transformation Semigroups Arising from Generalized Shifts

For a finite discrete topological space $X$ with at least two elements, a nonempty set $Gamma$, and a map $varphi:Gamma to Gamma$, $sigma_{varphi}:X^{Gamma} to X^{Gamma}$with $sigma_{varphi}((x_{alpha})_{alpha in Gamma})=(x_{varphi(alpha)})_{alpha in Gamma}$ (for $(x_{alpha})_{alpha in Gamma} in X^{Gamma}$) is a generalized shift. In this text for $mathcal{S} = {sigma_{varphi}:varphi in Gamma^{Gamma}}$ and $mathcal{H}={sigma_{varphi}:Gamma xrightarrow{varphi} Gamma$ is bijective$}$ we study proximal relations of transformation semigroups $(mathcal{S}, X^{Gamma})$ and $(mathcal{H}, X^{Gamma})$. Regarding proximal relation we prove: $$P(mathcal{S}, X^{Gamma}) = {((x_{alpha})_{alpha in Gamma},(y_{alpha})_{alpha in Gamma}) in X^{Gamma} times X^{Gamma} : exists beta in Gamma (x_{beta} = y_{beta})}$$and $P(mathcal{H}, X^{Gamma} ) subseteq {((x_{alpha})_{alpha in Gamma},(y_{alpha})_{alpha in Gamma}) in X^{Gamma} times X^{Gamma} : {beta in Gamma : x_{beta} = y_{beta}}$ is infinite$}$ $cup{($ $x,x) : x in mathcal{X}}$. Moreover, for infinite $Gamma$, both transformation semigroups $(mathcal{S}, X^{Gamma})$ and $(mathcal{H}, X^{Gamma})$ are regionally proximal, i.e., $Q(mathcal{S}, X^{Gamma}) = Q(mathcal{H}, X^{Gamma} ) = X^{Gamma} times X^{Gamma}$, also for sydetically proximal relation we have $L(mathcal{H}, X^{Gamma}) = {((x_{alpha})_{alpha in Gamma},(y_{alpha})_{alpha in Gamma}) in X^{Gamma} times X^{Gamma} : {gamma ∈ Gamma :$ $x_{gamma} neq y_{gamma}}$ is finite$}$.