# Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems

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## Abstract

**:**

## 1. Introduction and Preliminaries

- (1)
- If $t={x}_{i}$; $1\le i\le n$, then ${S}^{n}({x}_{i},{t}_{1},\dots ,{t}_{n}):={t}_{i}.$
- (2)
- If $t={x}_{i}$; $n<i$, then ${S}^{n}({x}_{i},{t}_{1},\dots ,{t}_{n}):={x}_{i}$.
- (3)
- If $t={f}_{i}({s}_{1},\dots ,{s}_{{n}_{i}})$, then${S}^{n}(t,{t}_{1},\dots ,{t}_{n}):={f}_{i}({S}^{n}({s}_{1},{t}_{1},\dots ,{t}_{n}),\dots ,{S}^{n}({s}_{{n}_{i}},{t}_{1},\dots ,{t}_{n})).$

- (1)
- $\widehat{\sigma}\left[{x}_{i}\right]:={x}_{i}\in X,$
- (2)
- $\widehat{\sigma}\left[{f}_{i}({t}_{1},\dots ,{t}_{{n}_{i}})\right]:={S}^{{n}_{i}}(\sigma \left({f}_{i}\right),\widehat{\sigma}\left[{t}_{1}\right],\dots ,\widehat{\sigma}\left[{t}_{{n}_{i}}\right])$ where $\widehat{\sigma}\left[{t}_{j}\right]$, $1\le j\le {n}_{i}$ are already defined.

- (1)
- If $B=\left\{{x}_{i}\right\}$; $1\le i\le n$, then ${\widehat{S}}^{n}(\left\{{x}_{i}\right\},{B}_{1},\dots ,{B}_{n}):={B}_{i}$.
- (2)
- If $B=\left\{{x}_{i}\right\}$; $n<i$, then ${\widehat{S}}^{n}(\left\{{x}_{i}\right\},{B}_{1},\dots ,{B}_{n}):=\left\{{x}_{i}\right\}$.
- (3)
- If $B=\left\{{f}_{i}({t}_{1},\dots ,{t}_{{n}_{i}})\right\}$, and suppose that ${\widehat{S}}^{n}(\left\{{t}_{k}\right\},{B}_{1},\dots ,{B}_{n})$ for all $k=1,\dots ,{n}_{i}$ are already defined, then ${\widehat{S}}^{n}(\left\{{f}_{i}({t}_{1},\dots ,{t}_{{n}_{i}})\right\},{B}_{1},\dots ,{B}_{n}):=\{{f}_{i}({r}_{1},\dots ,{r}_{{n}_{i}})\mid {r}_{k}\in {\widehat{S}}^{n}(\left\{{t}_{k}\right\},{B}_{1},\dots ,{B}_{n}),1\le k\le {n}_{i}\}$.
- (4)
- If B is an arbitrary nonsingleton subset of ${W}_{\tau}\left(X\right)$, then$${\widehat{S}}^{n}(B,{B}_{1},\dots ,{B}_{n}):=\bigcup _{b\in B}{\widehat{S}}^{n}(\left\{b\right\},{B}_{1},\dots ,{B}_{n}).$$

- (1)
- $\widehat{\sigma}[\varnothing ]:=\varnothing $,
- (2)
- $\widehat{\sigma}\left[\left\{{x}_{i}\right\}\right]:=\left\{{x}_{i}\right\}$ where ${x}_{i}$ is a variable from X,
- (3)
- $\widehat{\sigma}\left[\left\{{f}_{i}({t}_{1},\dots ,{t}_{{n}_{i}})\right\}\right]:={\widehat{S}}^{{n}_{i}}(\sigma \left({f}_{i}\right),\widehat{\sigma}\left[\left\{{t}_{1}\right\}\right],\dots ,\widehat{\sigma}\left[\left\{{t}_{{n}_{i}}\right\}\right])$ if $\widehat{\sigma}\left[\left\{{t}_{k}\right\}\right],1\le k\le {n}_{i}$ are already defined,
- (4)
- $\widehat{\sigma}\left[B\right]:={\displaystyle \bigcup _{b\in B}}\widehat{\sigma}\left[\left\{b\right\}\right]$ if B is an arbitrary nonsingleton subset of ${W}_{\tau}\left(X\right).$

## 2. Monomorphisms between Semigroups of Terms and Tree Languages

**Lemma**

**1.**

**Proof.**

- $\left\{{S}^{n}(t,{s}_{1},\dots ,{s}_{n})\right\}$
- $\hspace{1em}=\hspace{1em}\left\{{S}^{n}({f}_{i}({t}_{1},\dots ,{t}_{{n}_{i}}),{s}_{1},\dots ,{s}_{n})\right\}$
- $\hspace{1em}=\hspace{1em}\left\{{f}_{i}({S}^{n}({t}_{1},{s}_{1},\dots ,{s}_{n}),\dots ,{S}^{n}({t}_{{n}_{i}},{s}_{1},\dots ,{s}_{n}))\right\}$
- $\hspace{1em}=\hspace{1em}\{{f}_{i}({r}_{1},\dots ,{r}_{{n}_{i}})\mid {r}_{k}\in \left\{{S}^{n}({t}_{k},{s}_{1},\dots ,{s}_{n})\right\},1\le k\le {n}_{i}\}$
- $\hspace{1em}=\hspace{1em}\{{f}_{i}({r}_{1},\dots ,{r}_{{n}_{i}})\mid {r}_{k}\in {\widehat{S}}^{n}(\left\{{t}_{k}\right\},\left\{{s}_{1}\right\},\dots ,\left\{{s}_{n}\right\}),1\le k\le {n}_{i}\}$
- $\hspace{1em}=\hspace{1em}{\widehat{S}}^{n}(\left\{{f}_{i}({t}_{1},\dots ,{t}_{{n}_{i}})\right\},\left\{{s}_{1}\right\},\dots ,\left\{{s}_{{n}_{i}}\right\})$
- $\hspace{1em}=\hspace{1em}{\widehat{S}}^{n}(\left\{t\right\},\left\{{s}_{1}\right\},\dots ,\left\{{s}_{{n}_{i}}\right\}).$

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

- $\phi \left(({s}_{1},\dots ,{s}_{n}){\ast}_{g}({t}_{1},\dots ,{t}_{n})\right)$
- $\hspace{1em}=\hspace{1em}\phi \left(({S}^{n}({s}_{1},{t}_{1},\dots ,{t}_{n}),\dots ,{S}^{n}({s}_{n},{t}_{1},\dots ,{t}_{n}))\right)$
- $\hspace{1em}=\hspace{1em}(\left\{{S}^{n}({s}_{1},{t}_{1},\dots ,{t}_{n})\right\},\dots ,\left\{{S}^{n}({s}_{n},{t}_{1},\dots ,{t}_{n})\right\})$
- $\hspace{1em}=\hspace{1em}({\widehat{S}}^{n}(\left\{{s}_{1}\right\},\left\{{t}_{1}\right\},\dots ,\left\{{t}_{n}\right\}),\dots ,{\widehat{S}}^{n}(\left\{{s}_{n}\right\},\left\{{t}_{1}\right\},\dots ,\left\{{t}_{n}\right\}))$
- $\hspace{1em}=\hspace{1em}(\left\{{s}_{1}\right\},\dots ,\left\{{s}_{n}\right\}){\ast}_{g}(\left\{{t}_{1}\right\},\dots ,\left\{{t}_{n}\right\})$
- $\hspace{1em}=\hspace{1em}\phi \left(({s}_{1},\dots ,{s}_{n})\right){\otimes}_{g}\phi \left(({t}_{1},\dots ,{t}_{n})\right).$

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 3. The Left Seminearring of Non-Deterministic Generalized Hypersubstitutions

**Definition**

**1.**

**Proposition**

**6.**

**Theorem**

**1.**

**Proof.**

**Proposition**

**7.**

**Proof.**

$\left(\left({\sigma}_{1}{+}_{G}^{nd}{\sigma}_{2}\right){+}_{G}^{nd}{\sigma}_{3}\right)\left({f}_{i}\right)$ | = | ${\widehat{S}}^{{n}_{i}}({\sigma}_{3}\left({f}_{i}\right),\left({\sigma}_{1}{+}_{G}^{nd}{\sigma}_{2}\right)\left({f}_{i}\right),\dots ,\left({\sigma}_{1}{+}_{G}^{nd}{\sigma}_{2}\right)\left({f}_{i}\right))$ |

= | ${\widehat{S}}^{{n}_{i}}({\widehat{S}}^{{n}_{i}}({\sigma}_{3}\left({f}_{i}\right),{\sigma}_{2}\left({f}_{i}\right),\dots ,{\sigma}_{2}\left({f}_{i}\right)),{\sigma}_{1}\left({f}_{i}\right),\dots ,{\sigma}_{1}\left({f}_{i}\right))$ | |

= | ${\widehat{S}}^{{n}_{i}}(\left({\sigma}_{2}{+}_{G}^{nd}{\sigma}_{3}\right)\left({f}_{i}\right),{\sigma}_{1}\left({f}_{i}\right),\dots ,{\sigma}_{1}\left({f}_{i}\right))$ | |

= | $\left({\sigma}_{1}{+}_{G}^{nd}\left({\sigma}_{2}{+}_{G}^{nd}{\sigma}_{3}\right)\right)\left({f}_{i}\right).$ |

**Theorem**

**2.**

**Proof.**

$\left({\sigma}_{1}{\circ}_{G}^{nd}\left({\sigma}_{2}{+}_{G}^{nd}{\sigma}_{3}\right)\right)\left({f}_{i}\right)$ | = | ${\widehat{\sigma}}_{1}\left[\left({\sigma}_{2}{+}_{G}^{nd}{\sigma}_{3}\right)\left({f}_{i}\right)\right]$ |

= | ${\widehat{\sigma}}_{1}\left[{\widehat{S}}^{n}({\sigma}_{3}\left({f}_{i}\right),{\sigma}_{2}\left({f}_{i}\right),\dots ,{\sigma}_{2}\left({f}_{i}\right))\right]$ | |

= | ${\widehat{S}}^{n}({\widehat{\sigma}}_{1}\left[{\sigma}_{3}\left({f}_{i}\right)\right],{\widehat{\sigma}}_{1}\left[{\sigma}_{2}\left({f}_{i}\right)\right],\dots ,{\widehat{\sigma}}_{1}\left[{\sigma}_{2}\left({f}_{i}\right)\right])$ | |

= | ${\widehat{S}}^{n}(\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{3}\right)\left({f}_{i}\right),\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{2}\right)\left({f}_{i}\right),\dots ,\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{2}\right)\left({f}_{i}\right))$ | |

= | $\left(\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{2}\right){+}_{G}^{nd}\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{3}\right)\right)\left({f}_{i}\right).$ |

**Example**

**1.**

$\left(\left({\sigma}_{1}{+}_{G}^{nd}{\sigma}_{2}\right){\circ}_{G}^{nd}{\sigma}_{3}\right)\left(f\right)$ | = | ${\left({\sigma}_{1}{+}_{G}^{nd}{\sigma}_{2}\right)}^{\widehat{}}\left[\left\{f({x}_{2},{x}_{1})\right\}\right]$ |

= | ${\widehat{S}}^{2}(\left({\sigma}_{1}{+}_{G}^{nd}{\sigma}_{2}\right)\left(f\right),\left\{{x}_{2}\right\},\left\{{x}_{1}\right\})$ | |

= | ${\widehat{S}}^{2}(\{{x}_{4},f({x}_{3},f({x}_{1},f({x}_{2},{x}_{5})))\},\left\{{x}_{2}\right\},\left\{{x}_{1}\right\})$ | |

= | $\{{x}_{4},f({x}_{3},f({x}_{2},f({x}_{1},{x}_{5})))\}$ |

- $\left(\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{3}\right){+}_{G}^{nd}\left({\sigma}_{2}{\circ}_{G}^{nd}{\sigma}_{3}\right)\right)\left(f\right)$
- $\hspace{1em}=\hspace{1em}{\widehat{S}}^{2}(\left({\sigma}_{2}{\circ}_{G}^{nd}{\sigma}_{3}\right)\left(f\right),\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{3}\right)\left(f\right),\left({\sigma}_{1}{\circ}_{G}^{nd}{\sigma}_{3}\right)\left(f\right))$
- $\hspace{1em}=\hspace{1em}{\widehat{S}}^{2}(\{{x}_{4},f({x}_{1},{x}_{5})\},\left\{f({x}_{2},f({x}_{1},{x}_{5}))\right\},\left\{f({x}_{2},f({x}_{1},{x}_{5}))\right\})$
- $\hspace{1em}=\hspace{1em}\{{x}_{4},f(f({x}_{2},f({x}_{1},{x}_{5})),{x}_{5})\}$

**Definition**

**2.**

- (1)
- projection non-deterministic generalized hypersubstitution if the image $\sigma \left({f}_{i}\right)$ is a nonempty subset of X. Let ${P}_{G}^{nd}\left(\tau \right)$ be the set of all projection non-deterministic generalized hypersubstitutions of type τ.
- (2)
- a pre-non-deterministic generalized hypersubstitution if $\sigma \left({f}_{i}\right)$ is a nonempty subset of ${W}_{\tau}\left(X\right)\backslash X$. Let $Pr{e}_{G}^{nd}\left(\tau \right)$ be the set of all pre-non-deterministic generalized hypersubstitutions of type τ.

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**2.**

**Theorem**

**5.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**1.**

## 4. Representation of Menger Algebra with Infinitely Many Nullary Operations by n-Ary Functions

**Lemma**

**4.**

**Proof.**

${\lambda}_{{S}^{n}(t,{s}_{1},\dots ,{s}_{n})}({v}_{1},\dots ,{v}_{n})$ | = | ${S}^{n}({S}^{n}(t,{s}_{1},\dots ,{s}_{n}),{v}_{1},\dots ,{v}_{n})$ |

= | ${S}^{n}(t,{S}^{n}({s}_{1},{v}_{1},\dots ,{v}_{n}),\dots ,{S}^{n}({s}_{n},{v}_{1},\dots ,{v}_{n}))$ | |

= | ${\lambda}_{t}({S}^{n}({s}_{1},{v}_{1},\dots ,{v}_{n}),\dots ,{S}^{n}({s}_{n},{v}_{1},\dots ,{v}_{n}))$ | |

= | ${\lambda}_{t}({\lambda}_{{s}_{1}}({v}_{1},\dots ,{v}_{n}),\dots ,{\lambda}_{{s}_{n}}({v}_{1},\dots ,{v}_{n}))$ | |

= | $\mathcal{O}({\lambda}_{t},{\lambda}_{{s}_{1}},\dots ,{\lambda}_{{s}_{n}})({v}_{1},\dots ,{v}_{n}).$ |

**Lemma**

**5.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Example**

**2.**

${S}^{1}$ | ${x}_{1}$ | ${x}_{2}$ | $f({x}_{2},{x}_{3})$ | $f({x}_{4},f({x}_{5},{x}_{3}))$ |

${x}_{1}$ | ${x}_{1}$ | ${x}_{2}$ | $f({x}_{2},{x}_{3})$ | $f({x}_{4},f({x}_{5},{x}_{3}))$ |

${x}_{2}$ | ${x}_{2}$ | ${x}_{2}$ | ${x}_{2}$ | ${x}_{2}$ |

$f({x}_{2},{x}_{3})$ | $f({x}_{2},{x}_{3})$ | $f({x}_{2},{x}_{3})$ | $f({x}_{2},{x}_{3})$ | $f({x}_{2},{x}_{3})$ |

$f({x}_{4},f({x}_{5},{x}_{3}))$ | $f({x}_{4},f({x}_{5},{x}_{3}))$ | $f({x}_{4},f({x}_{5},{x}_{3}))$ | $f({x}_{4},f({x}_{5},{x}_{3}))$ | $f({x}_{4},f({x}_{5},{x}_{3}))$ |

$\mathcal{O}$ | ${\lambda}_{{x}_{1}}$ | ${\lambda}_{{x}_{2}}$ | ${\lambda}_{f({x}_{2},{x}_{3})}$ | ${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ |

${\lambda}_{{x}_{1}}$ | ${\lambda}_{{x}_{1}}$ | ${\lambda}_{{x}_{2}}$ | ${\lambda}_{f({x}_{2},{x}_{3})}$ | ${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ |

${\lambda}_{{x}_{2}}$ | ${\lambda}_{{x}_{2}}$ | ${\lambda}_{{x}_{2}}$ | ${\lambda}_{{x}_{2}}$ | ${\lambda}_{{x}_{2}}$ |

${\lambda}_{f({x}_{2},{x}_{3})}$ | ${\lambda}_{f({x}_{2},{x}_{3})}$ | ${\lambda}_{f({x}_{2},{x}_{3})}$ | ${\lambda}_{f({x}_{2},{x}_{3})}$ | ${\lambda}_{f({x}_{2},{x}_{3})}$ |

${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ | ${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ | ${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ | ${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ | ${\lambda}_{f({x}_{4},f({x}_{5},{x}_{3}))}$ |

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kumduang, T.; Leeratanavalee, S.
Semigroups of Terms, Tree Languages, Menger Algebra of *n*-Ary Functions and Their Embedding Theorems. *Symmetry* **2021**, *13*, 558.
https://doi.org/10.3390/sym13040558

**AMA Style**

Kumduang T, Leeratanavalee S.
Semigroups of Terms, Tree Languages, Menger Algebra of *n*-Ary Functions and Their Embedding Theorems. *Symmetry*. 2021; 13(4):558.
https://doi.org/10.3390/sym13040558

**Chicago/Turabian Style**

Kumduang, Thodsaporn, and Sorasak Leeratanavalee.
2021. "Semigroups of Terms, Tree Languages, Menger Algebra of *n*-Ary Functions and Their Embedding Theorems" *Symmetry* 13, no. 4: 558.
https://doi.org/10.3390/sym13040558