Can we define the normal set without $G$ being a group? Let $X$ be a set in $G$ and $G$ be a group. A normal set is a set $X$ for which $gxg⁻¹∈X$ for every $x∈X,g∈G$. It's just like the normality condition for subgroups, except that $X$ doesn't have to be a subgroup. 
I hvae two questions:
(1) Can we define the normal set without $G$ being a group?
(2) The normal vector to a surface is a vector perpendicular to it. Does there is a relation between this notion and the notion of normal set?
 A: Your definition of normal subsets $X$ of group $G$ is absolutely valid. One can easily see that a subset is normal iff it is the union of conjugacy classes (a conjugacy class of an element $x \in G$ is the set $\{g^{-1}xg: g \in G\}$, which is the equivalence class of $x$ under the equivalence relation conjugation). Conjugacy classes are important in groups for several reasons beyond the scope of this answer. And with them, one can actually  "construct" normal sets. Observe that complements $G-N$ of normal subgroups $N$ of $G$ are normal subsets.In a similar vein one can define the centralizer $C_G(X)=\{g \in G:g^{-1}xg=x$ for all $x \in X\} $ and normalizer of a set $X$, $N_G(X)=\{g \in G:g^{-1}Xg=X\}$.
A: A congruence in a monoid $M$ is an equivalence relation $\equiv$ in $M$ that is compatible with the operation of $M$:
$$
a \equiv b, a' \equiv b' \implies aa '\equiv bb'
$$
The quotient $\overline M = M\,/\equiv$ is then a monoid.
It is easy to prove that, when $G$ is a group and $\equiv$ is a congruence in $G$, the equivalence class of $1$ is a normal subgroup $N$ of $G$ and the equivalence classes are the cosets of $N$. Conversely, if $N$ is a normal subgroup of $G$, then the relation defined by $a \equiv b$ if $a^{-1}b \in N$ is a congruence relation in $G$ whose equivalence classes are the cosets of $N$.
I don't think there is much to say about congruences in a monoid in general because given a submonoid $N$ of a monoid $M$, the cosets of $N$ in $M$ are not necessarily disjoint and so do not define equivalence classes.
