Regular Languages
Definition
Let $\Sigma $ be an alphabet. A language $L \subset \str(\Sigma )$ is called regular if there exists a finite automaton that recognizes it.
Regular operations
Let $A, B \subset \str(\Sigma )$ be languages in $\Sigma $.
Union
The union (alternation) of $A$ and $B$ is, as usual, the set $A \cup B$.Concatenation
The concatenation of $A$ and $B$ is the set $\Set{xy}{x \in A \text{ and } y \in B}$, where $xy$ denotes length $\num{x}+\num{y}$ string which is the concatenation of $x$ and $y$Multi-concatenation
The star (Kleene star, multi-concatenation) of $A$ is the set\[ \Set{x \in \str(\Sigma )}{\exists k \geq 0, x = y_1y_2 \cdots y_k, y_i \in A}. \]
By this definition we do mean to include the empty string $\varnothing$ in $A^*$, regardless of $A$.Notation
We denote the alternation of $A$ and $B$ by $A \cup B$ as usual, but other notations include $A + B$, $A\mid B$, and $A \lor B$. We denote the concatenation of $A$ and $B$ by $AB$, but other notations include $A \circ B$. We denote the star of $A$ by $A^*$.