TITLE: What is a simple loop?
QUESTION [5 upvotes]: I'm asking here because no textbook or website that I know of gives a definition of the above mentioned term.
Since there's no obvious way (that I can think of) to define a normal subloop, I don't see how the definition of a simple group can be modified for the case of loops.
So - what is the definition of a simple loop? (And what is the motivation behind the definition?)

REPLY [3 votes]: Without referencing homomorphisms, a normal subloop K is one such that as sets:
$$\begin{align}
xK &= Kx \\
(xy)K &= x(yK) \\
K(xy) &= (Kx) y
\end{align}$$
Each of these can also be read as $\forall x, y, k \, \exists k'(x, y, k)$ such that $xk = k'x$, etc.  The first equation is just the normality condition for groups.  The second two assert that "next to" elements of the normal subgroup, associativity holds up to choosing new elements from the normal subloop.  This is enough to get Lagrange's theorem to hold, and for right and left cosets to be equal.