The concept of regularity in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann. It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees.
The concept of regularity for semigroups was inspired by John von Neumann's work on rings and later developed by Green.  David Rees is credited with initially suggesting applying the concept of regularity to semigroups, as noted in a footnote from Green's 1951 publication.