The road closure conjecture

There is a new approach that has been proposed recently for the study of certain regular semigroups. This new approach involves the study of finite primitive groups. Via this method, many problems in the context of regular semigroups translate into natural problems in finite primitive groups. We give some details concerning the relation between regular semigroups and finite primitive groups. In particular, we discuss the Road Closure Conjecture on finite primitive groups, which is strictly related to a possible classification of certain idempotent generated regular semigroups.