Pumping lemma and $L \subset \{a\}^*$ Let $L \subset \{a\}^*$ and $L$ satisfies pump lemma. Prove that $L$ is regular.
Please help me.
My an attempt:
Definition. A language $L$ of $A^∗$ is recognized by a monoid $M$ if there is a monoid morphism11 $f:A^* \rightarrow M$ and a subset $X$ of $M$ such that $f^{-1}(X)=L$.
Fact. A language is regular if and only if it is recognized by some finite monoid. 
Let $A = \{a\}$
Ok, Let $M = (\mathbb{N}, +, 0)  $ be a monoid. And let $h : A^* \rightarrow M$ be a homomorphism defined: $h(s) = |s|$. Now, we should find such $X$ that $h^{-1}[X]  = L$. So, let consider $h[L]$. h[L] is a set with a numbers which means length of words in $L$. So, Let $X = h[L] $.
It is important to observe that $h$ is injection- $A = \{a\}$. 
This proof is not used for pumping lemma. I have to prove that the lemma of pumping is necessary and sufficient here. Therefore, I ask for help in finding such evidence as well as check my proof.
 A: First let us define one useful notation: we say that a positive integer $w \in \mathbb{Z}$ belongs to L if $a^w \in L$. In other words, $w \in L \iff a^w \in L$.
We will prove that a language that satisfies pumping lemma is periodical starting from some point. In other words, there exist numbers $s, f \in \mathbb{N}$ and a set of residues $F \subset [0..f)$ modulo $f$ such that for every $w \ge s$ we have $w \in L \iff w \mod f \in F$. Such languages are clearly regular.
The pumping lemma states that there exists $p \ge 1$ such that every $w \in L$ with $w \ge p$ can be written as $w = x + y + z$ with the following conditions:


*

*$y \ge 1$

*$x + y \le p$

*for all $i \ge 0: x+iy+z \in L$


We can restate this:  
$$
w + i \cdot y \in L, y \le p
$$
In other words, starting from $w$ every number that is congruent to $w$ modulo $y$ belongs to $L$. That is already something periodic. We know that for every $w \in L$ there exists a corresponding $y_w \le p$ such that $w + i \cdot y_w \in L$. Let us take the least common multiple of all such $y_w$. Since all of them are less than $p$ there is only a finite number of distinct $y_w$ and hence their lcm is well-defined. Let us denote it $f$. It is the period we are looking for. Indeed, we know that if $w \in L$ then $w + f \in L$ (since $f$ is a multiple of $y_w$ by construction). Now in every residual class modulo f we can take the first number that belongs to $L$ (if there is at least one). Let us define $s$ as the maximum of all such numbers. Then it is clear that $w + f \in L$ and $w \ge s$ then $w \in L$. That is what we need: starting from $s$ the set of numbers $w \in L$ is periodic with period $f$.
