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 $\phi :\Gamma \to \Gamma$, ${\sigma }_{\phi }:X^{\Gamma }\to X^{\Gamma }$ with ${\sigma }_{\phi }((x_{\alpha }{)}_{\alpha \in \Gamma })=(x_{\phi (\alpha )}{)}_{\alpha \in \Gamma }$ (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 $\mathcal{H}=\{{\sigma }_{\psi }:\Gamma \stackrel{\psi }{\to }\Gamma$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S},X^{\Gamma })$ and $(\mathcal{H},X^{\Gamma })$. Regarding proximal relation we prove: \[P({\mathcal S},X^Γ)=\{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \existsβ\inΓ\:(x_β=y_β)\}\] 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 $L(\mathcal{H},X^{\Gamma })=\{((x_{\alpha }{)}_{\alpha \in \Gamma },(y_{\alpha }{)}_{\alpha \in \Gamma })\in X^{\Gamma }\times X^{\Gamma }:\{\gamma \in \Gamma :x_{\gamma }\ne y_{\gamma }\}$ is finite$\}$.

arXiv comment8 pages

References in corpus (1)

• On generalized shift transformation semigroups