Abstract:

Although the study of semigroups has begun since the beginning of the last century, the algebraic theory of semigroups, especially the algebraic theory of regular semigroups, did not obtain systematic and considerable developments until the establishment of a set of Green’s relations in 1951. This has in turn completely illustrated the effectivity of Green’s method led by the set of Green’s relations on the study of semigroups, in particular, regular semigroups. During the recent 40 years, in order to extend the reserch areas of semigroups starting from regular semigroups, a series of generalized Green’s relations have been established. In view of this, we will make a systematic survey to an origin and a kind of sequence of generalizations for Green’s relations.  We focus our survey mainly on the work of the Chinese people, of course, also relates to some of the work overseas.

Key words: Green’s relations;  generalized Green’s relations; abundant semigroups; r-ample ［l-ample］ semigroups; E(S) ［U］-semiabundant (or ample) semigroups