an infinite monoid that is not free monoid and does not contain any free monoid

Question

Let $H$ generated by some generators $H=\langle h_1, \ldots, h_n\rangle$ $(n\gt 1)$. My question is whether there exists any monoid $H$ such that $H$, is infinite and $H$ is not a free monoid and $H$ does not contain any free monoid?

Answer 1

The paper https://arxiv.org/abs/1912.01695 gives a finitely presented semigroup with zero satisfying $x^9=0$ for all $x$. Adjoining an identity would give a monoid. The paper is not published and in Russian. This was a famous question.

Answer 2

Let $M$ be the quotient of the free monoid $\{a,b\}^*$ by the relations $x^3y = yx^3 = x^3$, for all words $x$ and $y$. This is a monoid with zero in which $x^3 = 0$ is an identity. To prove it is infinite, consider the Prouhet-Thue-Morse infinite word. This infinite word is known to be cube-free, which means that it contains no factor of the form $uuu$, where $u$ is a nonempty word. It follows that each finite prefix of this infinite word corresponds to an element of $M$, and thus $M$ is infinite.