I am reading some notes as an introduction to automata and formal language. The author uses the following notation, when speaking of string concatenation: $\Sigma^{*} \times \Sigma^{*} \rightarrow \Sigma^{*}$
where I believe the cross means cartesian product of two sets.
However, reading up on the cartesian product on Wikipedia, it seems like the elements of $A \times B$ are ordered pairs written in cartesian form $(a,b)$ where $a\in A$ and $b\in B$. So I am not sure if this is what is meant in the notes.
What would be the meaning of $A \times A \rightarrow A$ ?
Thank you.
Yes, that is what was meant to be written in the notes.
Concatenation : $\Sigma^{*} \times \Sigma^{*} \rightarrow \Sigma^{*}$ means that concatenation is a function from $\Sigma^{*} \times \Sigma^{*}$ to $\Sigma^{*}$, i.e. concatenation takes two elements (in a particular order) from the set $\Sigma^{*}$ (viz. $2$ words), and returns another element of the set $\Sigma^{*}$ (viz. another word).