There are ten magmata of order two up to isomorphism. They are as follows:
- The constant semigroup
- The left zero band
- The right zero band
- The semilattice (chain)
- The group
- A nonassociative right rack
- A nonassociative left rack
- A nonassociative, nonrack right shelf
- A nonassociative, nonrack left shelf
- The NAND/NOR magma.
There are also ten bands of order three up to isomorphism. They are as follows:
- The left zero band
- The right zero band
- The chain
- The “V” semilattice
- A miscellaneous band with two left-absorbing elements,
- The opposite of the above,
- The left flip-flop monoid,
- The right flip-flop monoid.
To be clear, this is merely a coincidence, or at least there is no compelling fact I can think of underlying both numbers, and there is no particular reason to believe there is one, especially as the former has three left-right pairs and the latter has four.
However, this does not preclude the existence of a “nicest bijection” between the two, and it might be a fun exercise to decide which one that is. We clearly want the bijection to send as many left-right pairs as possible to left-right pairs, so we have a choice of 24 different “cores” that do this, and for each of these 24 “cores” we have a further 24 “shells” to choose, so we naively have 576 choices. However, we definitely want to exchange the left/right zero bands, so we actually only have 6 choices of “core”, and we definitely want to exchange the chains, so we actually only have 6 choices of “shell”, reducing the choices drastically down to 36.
Further, I think we should probably have the racks sent to the zeroes plus zero, as they’re the ones “closer” to being quasigroups in my opinion, leaving 12 choices. Also, I think should be exchanged with , as they both “sit between” the right and left zeroes in different senses, leaving only four choices.
I think it is probably best to exchange with , as they are the most “miscellaneous” pairs, leaving a final pair of bijections to choose between.
One of these exchanges and and the other exchanges it and , and I genuinely have no preference between this pair, as I feared.