On Finite free objects I asked examples of finite free objects. I got answer "The free idempotent semigroup satisfies $x^2=x$ for each element $x$. And I confirm it is finite if it is finitely generated." with reference to a paper.
If I understood correctly the concept of free object, then elements of free idempotent semigroup of three generators can be seen as strings like "abaca" with "non-repeating" property: do not count for example "abcbc" because has "bc" repeats. Well, on The longest string of none consecutive repeated pattern it is said that there is infinitely many strings of three letters with that property.
So either at least one of the answers is wrong, or then I have understood this wrong.
If given semigroup is finite, how many elements it has?