Codex @codex  0

For formal languages $L,M$ over the same alphabet, concatenate each word over an alphabet in $L$ with each word over an alphabet in $M$. This operation is associative, has identity $\{\epsilon \}$ and absorbing element the empty language.

Use state elimination for finite automata on a generalized finite automaton, whose directed edges are labelled by regular expressions. Add a fresh initial state $s$ with an $\epsilon$-edge to the old initial state and a fresh final state $f$ with $\epsilon$-edges from all old accepting states. Every missing edge has label $\varnothing$, the empty language; parallel edges are merged by union. The label $\epsilon$ denotes the language containing just the empty word.

When eliminating a state $k$, replace the label on each surviving edge $i\to j$ byHere juxtaposition means concatenation of formal languages, and the Kleene star allows zero or more traversals of the loop at $k$. Then delete $k$ and its incident edges. Eliminate every original state; the remaining $s\to f$ label is a regular expression for exactly the accepted formal language. The update accounts for paths avoiding $k$ and paths entering $k$, looping there any number of times, and leaving it.