Let $S$ be a semigroup. A subsemigroup $N$ of $S$ is said to be left-normal if $g N \subseteq N g$ for all $g \in S$ and it is said to be right-normal if $g N \supseteq N g$ 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$$ g N \subseteq N g = g g^{-1} N g \subseteq g N g^{-1} g = g N \text{.}$$ 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

	and

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

Bibliography

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.