State elimination for finite automata ($R_{ij}\leftarrow R_{ij}\cup R_{ik}{R}_{kk}^{\ast }{R}_{kj}$)

To remove an internal state $k$, replace each surviving label by $R_{ij}\cup R_{ik}(R_{kk}{)}^{\ast }{R}_{kj}$. The added term accounts for entering $k$, traversing its loop any number of times, and leaving it. Fresh initial and final states allow all old states to be eliminated, producing a regular expression for the same language.