Wikipedia has the following definition of the pumping lemma for regular langauges...
Let $L$ be a regular language. Then there exists an integer $p$ ≥ 1 depending only on $L$ such that every string $w$ in $L$ of length at least $p$ ($p$ is called the "pumping length") can be written as $w$ = $xyz$ (i.e., $w$ can be divided into three substrings), satisfying the following conditions:
- |$y$| ≥ 1
- |$xy$| ≤ $p$
- for all $i$ ≥ 0, $xy^iz$ ∈ $L$
I do not see how this is satisfied for a simple finite regular language. If I have an alphabet of {$a,b$} and regular expression $ab$ then $L$ consists of just the one word which is $a$ followed by $b$. I now want to see if my regular language satisfies the pumping lemma...
As nothing repeats in my regular expression the value of $y$ must be empty so that condition 3 is satisifed for all $i$. But if so then it fails condition 1 which says $y$ must be at least 1 in length!
If instead I let $y$ be either $a$, $b$ or $ab$ then it will satisfy condition 1 but fail condition 3 because it never actually repeats itself.
I am obviously missing something mind blowingly obvious. Which is?