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 $Γ$, and a map $\varphi:Γ\toΓ$, $σ_\varphi:X^Γ\to X^Γ$ with $σ_\varphi((x_α)_{α\inΓ})= (x_{\varphi(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is a generalized shift. In this text for $\mathcal{S}=\{σ_ψ:ψ\inΓ^Γ\}$ and $\mathcal{H}=\{σ_ψ: Γ\mathop{\rightarrow}\limits^ψΓ$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S},X^Γ)$ and $(\mathcal{H},X^Γ)$. 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 $P({\mathcal H},X^Γ)\subseteq \{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \{β\inΓ:x_β=y_β\}$ is infinite~$\}\cup\{ (x,x):x\in \mathcal{X}\}$. \\ Moreover, for infinite $Γ$, both transformation semigroups $({\mathcal S},X^Γ)$ and $({\mathcal H},X^Γ)$ are regionally proximal, i.e., $Q({\mathcal S},X^Γ)=Q({\mathcal H},X^Γ)=X^Γ\times X^Γ$, also for sydetically proximal relation we have $L({\mathcal H},X^Γ)=\{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \{γ\inΓ:x_γ\neq y_γ\}$ is finite$\}$.

Theorems & Definitions (30)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1
• proof