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?