Suppose $(M,*)$ is a magma, that is, just a set with a binary operation with no conditions imposed, and let $s$ be an element of $M$. Also, let $n$ be an integer greater than or equal to $2$. An $n$-th power of $s$, (note the indefinite article), is the value of an expression $s*...*s$, where there are parentheses in that expression. So, for example, $s*(s*s)$ and $(s*s)*s$ are the two (not necessarily distinct) 3rd powers of $s$, and $(s*s)*(s*(s*s))$ is a 5th power of $s$. And of course, $s*s$ is the sole second power of $s$. My question is, if $s*s=s$, then does all $n$-th powers of $s$ equal $s$? Note, we are not automatically assuming that the magma is associative.
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1$\begingroup$ Have you tried to prove this yourself? This is a straightforward induction. $\endgroup$– Noah SchweberCommented Oct 2, 2023 at 0:38
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Observe that the collection of powers of $s$ can be recursively defined as the smallest collection such that:
- $s$ is a power of $s$
- If $a$ and $b$ are powers of $s$ then $a*b$ is a power of $s$.
It is then straightforward to do induction over this recursive definition to show that if $s*s=s$ then every power of $s$ equals $s$.
