View source for Concatenation

''catenation''

For [[word]]s over some [[alphabet]], the word obtained by taking the symbols of each word in order: if $x = (x_1,\ldots,x_m)$ and $y = (y_1,\ldots,y_n)$ then the concatenation $xy$ is the word $(x_1,\ldots,x_m,y_1,\ldots,y_n)$: the notations $x|y$, $x \cdot y$ are also used.  Denoting the empty word by $\lambda$, we have $x \lambda = \lambda x = x$.

For [[Formal language|language]]s (sets of words), the concatenation language is the set of concatenations
$$
L_1 L_2 = \{ x y : x \in L_1,\, y \in L_2 \} \ .
$$

Concatenation is associative, and defines a [[semi-group]] structure on the set of words over an alphabet.

{{TEX|done}}