On proximal relations in transformation semigroups arising from generalized shifts
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866916095480299520 |
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| author | Shirazi, Fatemah Ayatollah Zadeh Fallahpour, Amir Mardanbeigi, Mohammad Reza Ahmadabadi, Zahra Nili |
| author_facet | Shirazi, Fatemah Ayatollah Zadeh Fallahpour, Amir Mardanbeigi, Mohammad Reza Ahmadabadi, Zahra Nili |
| contents | For a finite discrete topological space $X$ with at least two elements, a nonempty set $Γ$, and a map $φ:Γ\toΓ$, $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})= (x_{φ(α)})_{α\inΓ}$ (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$\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_13100 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On proximal relations in transformation semigroups arising from generalized shifts Shirazi, Fatemah Ayatollah Zadeh Fallahpour, Amir Mardanbeigi, Mohammad Reza Ahmadabadi, Zahra Nili Dynamical Systems 54H20 For a finite discrete topological space $X$ with at least two elements, a nonempty set $Γ$, and a map $φ:Γ\toΓ$, $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})= (x_{φ(α)})_{α\inΓ}$ (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$\}$. |
| title | On proximal relations in transformation semigroups arising from generalized shifts |
| topic | Dynamical Systems 54H20 |
| url | https://arxiv.org/abs/1909.13100 |