On proximal relations in transformation semigroups arising from generalized shifts

TL;DR

This paper investigates proximal relations in transformation semigroups generated by generalized shifts on finite discrete spaces, revealing their structure and properties, especially for infinite index sets.

Contribution

It characterizes proximal and regionally proximal relations in semigroups of generalized shifts, extending understanding of their dynamical behavior.

Findings

Proximal relation in semigroup $\\mathcal{S}$ includes pairs sharing at least one coordinate.

In semigroup $\\mathcal{H}$, proximal pairs have infinitely many matching coordinates or are identical.

Both semigroups are regionally proximal when the index set is infinite.

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$\}$.