On proximal relations in transformation semigroups arising from generalized shifts

Abstract

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 $\mathcal{S}=\{\sigma_\psi:\psi\in\Gamma^\Gamma\}$ and $\mathcal{H}=\{\sigma_\psi: \Gamma\mathop{\rightarrow}\limits^{\psi}\Gamma$ 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 $L({\mathcal H},X^\Gamma)=\{((x_\alpha)_{\alpha\in\Gamma},(y_\alpha)_{\alpha\in\Gamma}) \in X^\Gamma\times X^\Gamma: \{\gamma\in\Gamma:x_\gamma\neq y_\gamma\}$ is finite$\}$.