# Are there 45 unital magmas with three elements (up to isomorphism)?

How many unital magmas (magma with an identity element) with three elements are there (up to isomorphism)?

My approach:

• List out all of the possible 2x2 multiplication tables for the two non-identity elements. There are $$3^{4} = 81$$ of these.

• Extend these 81 tables by adding in the rows and columns for the identity element.

• Manually identify isomorphisms, by re-labelling the two non-identity elements.

• I found that of the 81 tables, there are 36 isomorphic pairs of tables, plus 9 more tables where re-labelling simply gives the same table again.

• Thus, I conclude that, up to isomorphism, there are $$36 + 9 = 45$$ unital magmas with three elements.

Is this correct? And is there a more efficient approach to obtaining the solution?

This is correct. I would phrase this in terms of group actions, although it sounds like my argument is pretty much the same as yours. Let the underlying set be $$M=\{1,x,y\}$$. Every unital magma with $$3$$ elements is isomorphic to a unital magma with underlying set $$M$$ and $$1$$ being the unit by appropriately labeling the elements. The number of unital magma structures on $$M$$ is the number of maps $$\{x,y\}\times\{x,y\}\rightarrow M$$ (cause the rest of the multiplication is determined by the unit axiom and there are no further restrictions), of which there are $$3^{2\cdot2}=81$$ many. To determine how many of these are isomorphic, note that an isomorphism of unital magmas necessarily preserves the unit, so the only possible isomorphism is always given by the involution $$\tau\colon M\rightarrow M$$ that fixes $$1$$ and switches $$x$$ and $$y$$ (and every multiplication determines one that is isomorphic to it via $$\tau$$ by transport of structure, i.e. $$m\mapsto\tau\circ m\circ(\tau\times\tau)$$).
Thus, we obtain that $$\mathbb{Z}/2\mathbb{Z}$$ acts on the set of unital magma structures on $$M$$ and the isomorphism classes of such structures are precisely the orbits of this action. The condition that a unital magma structure is a fixed point of this action is equivalent to $$\tau(x^2)=y^2$$ and $$\tau(xy)=yx$$, so two of these products can be arbitrarily specified and the other two are then uniquely determined, meaning there are exactly $$3^2=9$$ such unital magma structures. This yields the overall number of orbits as $$9+\frac{81-9}{2}=45$$.