User:Jshflynn/Definition:Concatenation

Definition

Let $\Sigma$ be an alphabet and let $x$ and $y$ be words over $\Sigma$.

Then the concatenation of $x$ with $y$ is denoted $xy$ in the literature (though will usually be given a symbol such as $\circ$ on this site).

We define concatenation with the empty word as follows:

$x \circ \lambda = \lambda \circ x = x$

And in other cases:

$

(x \circ y)_i = \begin{cases} x_i & \text{if }1 \le i \le \operatorname{len}(x) \\ y_{i-\operatorname{len}(x)} & \text{if }\operatorname{len}(x)< i \le \operatorname{len}(x)+\operatorname{len}(y) \end{cases} $