one-sided normality of subsemigroup

Let $S$ be a semigroup. A subsemigroup $N$ of $S$ is said to beleft-normal if $gN\subseteq Ng$ for all $g\in S$ andit 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 automaticallynormal, 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-normalizersare 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

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

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

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

It may also be observed that the combination ‘left normal’ in semigrouptheory frequently occurs as part of the phrase ‘left normal band’,but in that case the etymology rather seems to be that ‘left’ qualifiesthe phrase ‘normal band’.