Power associative magma

Question

I’m looking for a magma with specific properties:
Requirements:
1.Power Associative(of course, I want it to not be alternative or similar).
2.Invertibility and identity element.
Preferences(In order of priority):
1.finite. 2.well known and or with a recognizabled name.
3.Simple to explain and construct.

When I first searched it online, I’ve found the sedenenions and the similar, which were way too complex for my wanting.
Any help will be appreciated.

Answer 1

I don't think it has a name, but here's a three element example (which is even commutative and has unique inverses) - note that since every two-element magma with an identity is alternative, this is the smallest example possible. (See the bottom of this answer for a more "symmetric" example in which however inverses are not unique.)

The elements of our magma $M$ are $e,x,y$. Unsurprisingly $e$ will be the identity; the rest of our multiplication table is given by $$xx=e, yy=e, xy=yx=x.$$ (I'm using concatenation for the magma operation for simplicity).

We trivially have an identity and unique inverses, and left alternativity fails since $$(xx)y=ey=y\color{red}{\not=}e=xx=x(xy)$$ (right alternativity fails similarly). To see that $M$ is power associative, simply note that each element of $M$ generates a fully associative submagma. And to top it off, $M$ is commmutative, which we didn't even ask for initially.

---

Let me say a bit about the general idea behind this answer. This was the following, especially the first bulletpoint:

Suppose we have a magma $M$ and submagmas $A_i$ ($i\in I$) such that $M=\bigcup_{i\in I}A_i$. Then:

• If each $A_i$ is associative, $M$ is power-associative.
• If there is some $e\in\bigcap_{i\in I}A_i$ which each $A_i$ thinks is the identity element then $e$ is in fact the identity in $M$.
• If the above bulletpoint holds and additionally each $A_i$ has inverses, then $M$ has inverses - and the inverse of an element of $A_i$ in the sense of $M$ is the inverse of that element in the sense of $A_i$.

This suggests a general way to build power associative magmas with identities and inverses. What we've done above is do this with two "components," each of which is in fact a group - namely, the "$x$-part" and "$y$-part" are both copies of $\mathbb{Z}/2\mathbb{Z}$. After determining our "components," and deciding how exactly they overlap (we want them to have a common identity, and for simplicity we might as well make them only have their identity elements in common), all we have to do is determine how elements from different "components" multiply.

But now we're almost home free: any decision we make here will give us a magma which is power associative and has an identity and inverses. So we just play around a bit and find a choice which makes things nice.

In a bit more detail, having decided to build a magma by "gluing two copies of $\mathbb{Z}/2\mathbb{Z}$ together at the identity," we have two decisions to make (letting $x$ and $y$ be the non-identity elements of our magma):

• Which of $x,y,e$ is $xy$?

• Which of $x,y,e$ is $yx$?

The most obvious alternative (hehe) to what we've done above is to declare $xy=yx=e$. The resulting magma is more "symmetric" - the map switching $x$ and $y$ is a magma automorphism - and is still a commutative power associative non-alternative magma with identity and inverses; however, it does not have unique inverses.