The concept of regularity in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann.
[3] 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 term inversive semigroup (French: demi-groupe inversif) was historically used as synonym in the papers of Gabriel Thierrin (a student of Paul Dubreil) in the 1950s,[4][5] and it is still used occasionally.
[8] The set of inverses (in the above sense) of an element a in an arbitrary semigroup S is denoted by V(a).
[9] Thus, another way of expressing definition (2) above is to say that in a regular semigroup, V(a) is nonempty, for every a in S. The product of any element a with any b in V(a) is always idempotent: abab = ab, since aba = a.
Recall that the principal ideals of a semigroup S are defined in terms of S1, the semigroup with identity adjoined; this is to ensure that an element a belongs to the principal right, left and two-sided ideals which it generates.
In a regular semigroup S, however, an element a = axa automatically belongs to these ideals, without recourse to adjoining an identity.