# free semigroup with involution

Let $X,{X}^{\u2021}$ be two disjoint sets in bijective^{} correspondence given by the map ${}^{\u2021}:X\to {X}^{\u2021}$. Denote by $Y=X\coprod {X}^{\u2021}$ (here we use $\coprod $ instead of $\cup $ to remind that the union is actually a disjoint union^{}) and by ${Y}^{+}$ the free semigroup^{} on $Y$. We can extend the map ${}^{\u2021}$ to an involution ${}^{\u2021}:{Y}^{+}\to {Y}^{+}$ on ${Y}^{+}$ in the following way: given $w\in {Y}^{+}$, we have $w={w}_{1}{w}_{2}\mathrm{\dots}{w}_{k}$ for some letters ${w}_{i}\in Y$; then we define

$${w}^{\u2021}={w}_{k}^{\u2021}{w}_{k-1}^{\u2021}\mathrm{\dots}{w}_{2}^{\u2021}{w}_{1}^{\u2021}.$$ |

It is easily verified that this is the unique way to extend ${}^{\u2021}$ to an involution on $Y$. Thus, the semigroup^{} ${(X\coprod {X}^{\u2021})}^{+}$ with the involution $\u2021$ is a semigroup with involution. Moreover, it is the *free semigroup with involution* on $X$, in the sense that it solves the following universal problem: given a semigroup with involution $S$ and a map $\mathrm{\Phi}:X\to S$, a semigroup homomorphism $\overline{\mathrm{\Phi}}:{(X\coprod {X}^{\u2021})}^{+}\to S$ exists such that the following diagram commutes:

$$\text{xymatrix}\mathrm{\&}X\text{ar}{[r]}^{\iota}\text{ar}{[d]}_{\mathrm{\Phi}}\mathrm{\&}{(X\coprod {X}^{\u2021})}^{+}\text{ar}{[dl]}^{\overline{\mathrm{\Phi}}}\mathrm{\&}S\mathrm{\&}$$ |

where $\iota :X\to {(X\coprod {X}^{\u2021})}^{+}$ is the inclusion map^{}. It is well known from universal algebra^{} that ${(X\coprod {X}^{\u2021})}^{+}$ is unique up to isomorphisms^{}.

If we use ${Y}^{*}$ instead of ${Y}^{+}$, where ${Y}^{*}={Y}^{+}\cup \{\epsilon \}$ and $\epsilon $ is the empty word^{} (i.e. the identity^{} of the monoid ${Y}^{*}$), we obtain a monoid with involution ${(X\coprod {X}^{\u2021})}^{*}$ that is the *free monoid with involution* on $X$.

Title | free semigroup with involution |
---|---|

Canonical name | FreeSemigroupWithInvolution |

Date of creation | 2013-03-22 16:11:30 |

Last modified on | 2013-03-22 16:11:30 |

Owner | Mazzu (14365) |

Last modified by | Mazzu (14365) |

Numerical id | 8 |

Author | Mazzu (14365) |

Entry type | Example |

Classification | msc 20M10 |

Defines | free semigroup with involution |

Defines | free monoid with involution |