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

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

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

$S=\Biggl{\{}\begin{pmatrix}k&m\\ 0&1\end{pmatrix}\Biggm{|}k,m\in\mathbb{Z}\Biggr{\}}\qquad\text{and}\qquad N=% \Biggl{\{}\begin{pmatrix}1&n\\ 0&1\end{pmatrix}\Biggm{|}n\in\mathbb{Z}\Biggr{\}}\text{,}$

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

	$\displaystyle\begin{pmatrix}k&m\\ 0&1\end{pmatrix}\begin{pmatrix}1&n\\ 0&1\end{pmatrix}=$	$\displaystyle\begin{pmatrix}k&kn+m\\ 0&1\end{pmatrix}\quad\text{and}$
	$\displaystyle\begin{pmatrix}1&n\\ 0&1\end{pmatrix}\begin{pmatrix}k&m\\ 0&1\end{pmatrix}=$	$\displaystyle\begin{pmatrix}k&n+m\\ 0&1\end{pmatrix}$

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

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