Let $H= \langle h_1, \ldots , h_n \rangle$ ($n>1$) be an infinite monoid that is not a free monoid. Does $H$ contain an isomorphic copy of a free monoid as a submonoid?
EDIT. It is a natural question. The corresponding question for groups is the Burnside problem.
Best Answer
Let $h$ be an element of $H$. If the submonoid generated by $h$ is not free, then there exist two integers $n \geqslant 0$ and $p > 0$ such that $h^n = h^{n + p}$. Thus $H$ is a torsion monoid. There exists a two-generated infinite monoid which is torsion, thus the answer to your question is negative. Consider for instance 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 $M$ in which every $x^3 = 0$ 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.
The corresponding problem for groups is the famous Burnside problem which also has a negative, but difficult, answer, first given by Golod and Shafarevich.