Looking for examples of finite loops and monoids

Question

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms.

Does anyone know some ressources for this? I already used Wikipedia and Google but the problem I encountered was that I could not find any examples that were not groups.

Answer 1

Several examples of finite loops (quasigroups with left and right identity) of small order with some additional properties can be found here.

Another source showing several examples of finite loops Cayley tables.

A third example of very group-like loop which is not a group, answer to a related Math.SE question.

Again, in the absence of associativity, inverses may be defined in a more general way: this is the case of inverse property loops. The smallest example of such loop which is not a group has order $7$. This latter is, to me, a brilliant example of what may happen if we omit some of the group axioms (associativity, in the case of loops).

Answer 2

Another example. If $\mathbb O$ is an algebra of octonions with a standard basis $e_0,\ldots,e_7$, then $L=\{\pm e_0,\pm e_1,\ldots,\pm e_7\}$ is a finite non-associative and non-abelian loop with elegant law of composition (Fano plane).