 Length of a word over an alphabet

The length of a word over an alphabet is the number of symbol positions in that finite string, counting repetitions. Thus $\ell (abba)=4$ and $\ell (\epsilon )=0$ for the empty word. Concatenation satisfies $\ell (uv)=\ell (u)+\ell (v)$. This differs from the word length of a group element, which minimizes over representing words using a chosen generating set of a group.