Green's relations on a connected algebraic monoid†

Abstract

Abstract Let S be a connected algebraic monoid with group of units G and idempotent set E(S). Let e f ϵ E(S),a,b ϵ S b regular. Then 1) if and only if x-1 ex = f for some x ϵ G.2) if and only if xay = b for some x y ϵ G. 3) if and only if ax = b for some x ϵ G. 4) if and only if ya = b for some y ϵ G. 5) if and only if ex = xe = a for some x ϵ G. 6) if and only if e′ ⩾ f for some e′ϵE(S) with . 7) bub = b for som u ϵ G. Several other related results are obtained. †This research was partially supported by NSF Grant MCS 8025597 †This research was partially supported by NSF Grant MCS 8025597 Notes †This research was partially supported by NSF Grant MCS 8025597