# Does 2nd power idempotency imply all nth powers idempotency?

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.

• Have you tried to prove this yourself? This is a straightforward induction. Commented Oct 2, 2023 at 0:38

Observe that the collection of powers of $$s$$ can be recursively defined as the smallest collection such that:
1. $$s$$ is a power of $$s$$
2. 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 $$s$$.