Examples of loops which have two-sided inverses.

Question

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are there commutative finite loops?

Answer 1

See also the Parker Loop which is a finite loop of order $2^{13}$ related to the binary Golay code, $M_{24}$ (largest sporadic Mathieu group), Conway's construction of the Monster group, etc.

Answer 2

I think most of your questions are answered by looking at Moufang loops.

A loop in which the left and right inverse agree (a loop with two-sided inverses) is called an IP-loop. Sometimes people replace a loop by an isotope, which basically scrambles and relabels the multiplication table (apply a row and column permutation, and a permutation of the underlying set). For groups, that would basically be crazy, but loops are not terribly messed up by such an operation.

A loop is a Moufang loop iff every isotope has two-sided inverses.

Non-associative, commutative, Moufang loops have order a multiple of 81, and there are two non-isomorphic such loops. They were constructed by M. Hall Jr.

Answer 3

I recently saw an example from this preprint that I think is pretty neat. It's an abelian loop of infinite order.

Let $S$ be the set consisting of $1$ and all odd prime numbers. Define an operation $\cdot$ on $S$ where $a \cdot b$ gives the smallest element of $S$ strictly larger than $\lvert a - b \rvert$.

Clearly this operation is commutative and closed. The identity is given by $1$ and every element is its own inverse.