A semigroup $S$ is called *monogenic* if $S$ is generated by some element $a$ (which is unique if $S$ is not a group) in the sense that $S=\{a^n:n\in\mathbb N\}$.

Observe that each mongenic group is finite cyclic. It is known that each subsemigroup of a monogenic group is a cyclic group.

On the other hand, a subsemigroup of a finite monogenic semigroup need not be monogenic. The simplest example is the subsemigroup $\{a^2,a^3,a^4=a^5\}$ of the monogenic semigroup $\{a,a^2,a^3,a^4=a^5\}$.

Let us call a semigroup $S$ *submonogenic* if it is isomorphic to a subsemigroup of a monogenic semigroup.

**Question 1.** *Is there any reasonable characterization of (finite) submonogenic semigroups?*

It is clear that each finite submonogenic semigroup $S$ has the following properties:

(1) $S$ is commutative;

(2) $S$ has a unique idempotent;

(3) the minimal ideal $I$ of $S$ is a monogenic group;

(4) for any $a,x,y\in S$ the equality $ax=ay\notin I$ implies $x=y$;

(5) for any $n\in\mathbb N$ and $x,y\in S$ the equality $x^n=y^n\notin I$ implies $x=y$.

**Question 2.** *Is each finite semigroup $S$ satisfying the conditions (1)--(3) submonogenic?*

**Question 2'** (added after appearing Mark Sapir's Counterexample to Question 2). *Is each finite semigroup $S$ satisfying the conditions (1)--(5) submonogenic?*

**Question 3.** *Is any reasonable classification of finite submonogenic semigroups?*