Normal Submagma?

Question

Is there a definition of normal submagma? visit https://en.wikipedia.org/wiki/Magma_(algebra)

For normal sub-quasi-group I found two:

1. A sub-quasi-group $H$ is called normal if there exists a normal congruence $\theta$ such that $H$ coincides with one of the congruence classes.
2. A sub-quasi-group $H$ of a quasi-group $(Q,.)$ is called normal if and only if $x.H=H.x$, $(x.y).H=x.(y.H)$ and $H.(x.y)=(H.x).y$, for all $x,y\in Q$.

If this is true and the two definitions are equivalent, then how to prove that the first definition gives the second one?

Answer 1

There is a definition of normal subset (of a semigroup) at page 24 of A Survey of Binary Systems by R.H. Bruck:

If $\theta$ is a homomorphism of a semigroup $S$ into a groupoid, the image $S\theta=S'$ is also a semigroup. If $s'$ is an element of $S'$, let $K=s'\theta^{-1}$ be the inverse image of $s'$; that is, $K$ is the set of all $s$ in $S$ such that $s\theta=s'$. Clearly $K$ has the following properties:

1. if $k\in K$, $x\in S$ and $xk\in K$, then $xK\subset K$;

2. if $k\in K$, $x\in S$ and $kx\in K$, then $Kx\subset K$;

3. if $k\in K$, $x,y\in S$ and $xky\in K$, then $xKy\subset K$;

A non-empty subset $K$ of $S$ with properties 1., 2., 3. is called a normal subset.

This is followed by a reflection about why the study of normal subsets (subgroupoids) is less interesting than normal subsemigroups (or subgroups):

To a given normal subset $K$ of $S$ there may correspond two (or more) distinct homomorphisms of $S$ with $K$ as an inverse image; herein resides, perhaps, the relative poverty of the theory.