When dealing with this question, I came to realize that I'm not sure how sensible it is to define prefixes for regular expressions. So far I've run into two sorts of prefix definitions, one for strings or words, and one for languages:
Let $\Sigma$ be an alphabet. The prefix of a word or a string $x\in\Sigma^\ast$, the Kleene- or concatenation closure over the alphabet, is defined as follows: $$ \operatorname{pre} x = \{x[1:i] \mid i \in\{1,\ldots, |x|\}\}, $$ where the left associative slice operator $(\cdot)[m:n]$ inclusively returns the characters between the indices $m$ and $n$ in its argument.
The prefix language $\operatorname{pre}\mathcal L$ is then defined using words prefixes: $$ \operatorname{pre}\mathcal L = \{ x \in \operatorname{pre} y \mid y \in \mathcal L \}\,. $$
Now this is all fine and good, but if one wished to check whether a certain type of language (such as the prefix one) was regular, one would have to come up with a regular expression for each of the 6 types of languages present in the recursive definition of regular languages. But how might those be defined?
The problem is, for a regular expression such as $\alpha\beta(\gamma\delta)\epsilon \in\operatorname{RE}(\Sigma)$, the set of regular expressions that can be generated using the alphabet $\Sigma$, we can't just take an arbitrary amount of characters off the end of it. This is because it might result in a situation like $\alpha\beta(\gamma$, where the the left parenthesis has been left unmatched by a right one, therefore making this regular expression invalid: it cannot generate a regular language based on the definition.
This has left me wondering, how might prefixes of regular expressions be defined sensibly, or is that even necessary or useful?