# one-sided normality of subsemigroup

Let $S$ be a semigroup^{}. A subsemigroup $N$ of $S$ is said to be
*left-normal* if $gN\subseteq Ng$ for all $g\in S$ and
it is said to be *right-normal* if $gN\supseteq Ng$ for all
$g\in S$.
One may similarly define *left-normalizers*

$${\mathrm{LN}}_{S}(N):=\{g\in S\mid gN\subseteq Ng\}$$ |

and *right-normalizers*

$${\mathrm{RN}}_{S}(N):=\{g\in S\mid Ng\subseteq gN\}\text{.}$$ |

A left-normal subgroup^{} $N$ of a *group* $S$ is automatically
normal, since

$$gN\subseteq Ng=g{g}^{-1}Ng\subseteq gN{g}^{-1}g=gN\text{.}$$ |

In is similarly shown for general $S$ and $N$ that if some $g\in {\mathrm{LN}}_{S}(N)$ has an inverse^{} ${g}^{-1}$ then ${g}^{-1}\in {\mathrm{RN}}_{S}(N)$ and vice versa. Left- and right-normalizers
are always closed under^{} multiplication (hence subsemigroups)
and contain the identity element^{} of $S$ if there is one.

An example of a left-normal but not right-normal $N\subseteq S$ can be constructed using matrices under multiplication, if one takes

$$S=\left\{\left(\begin{array}{cc}\hfill k\hfill & \hfill m\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\right|k,m\in \mathbb{Z}\}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}N=\left\{\left(\begin{array}{cc}\hfill 1\hfill & \hfill n\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\right|n\in \mathbb{Z}\}\text{,}$$ |

where one may note that $N$ is a group and $S$ is a monoid. Since

$\left(\begin{array}{cc}\hfill k\hfill & \hfill m\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 1\hfill & \hfill n\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)=$ | $\left(\begin{array}{cc}\hfill k\hfill & \hfill kn+m\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\mathit{\hspace{1em}}\text{and}$ | ||

$\left(\begin{array}{cc}\hfill 1\hfill & \hfill n\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cc}\hfill k\hfill & \hfill m\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)=$ | $\left(\begin{array}{cc}\hfill k\hfill & \hfill n+m\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$ |

it follows that $gN\subseteq Ng$ for all $g\in S$, with proper inclusion (http://planetmath.org/ProperSubset) when $k\ne \pm 1$.

The definition of left and is somewhat arbitrary in the choice of whether to call something the or left form. A reference supporting the choice documented here is:

## References

- 1 Karl Heinrich Hofmann and Michael Mislove: The centralizing theorem for left normal groups of units in compact monoids, Semigroup Forum 3 (1971/72), no. 1, 31–42.

It may also be observed that the combination^{} ‘left normal’ in semigroup
theory frequently occurs as part of the phrase ‘left normal band’,
but in that case the etymology rather seems to be that ‘left’ qualifies
the phrase ‘normal band’.

Title | one-sided normality of subsemigroup |
---|---|

Canonical name | OnesidedNormalityOfSubsemigroup |

Date of creation | 2013-03-22 16:10:41 |

Last modified on | 2013-03-22 16:10:41 |

Owner | lars_h (9802) |

Last modified by | lars_h (9802) |

Numerical id | 6 |

Author | lars_h (9802) |

Entry type | Definition |

Classification | msc 20A05 |

Defines | left-normal |

Defines | right-normal |

Defines | left-normalizer |

Defines | right-normalizer |