Regular Simple ω²-Semigroups with ω_{d, d′}-Type $\mathcal{D}$ Relation

To obtain the characterization of regular simple ω²-semigroups, the regular simple ω²-semigroups with ω_{d, d′}-type $\mathcal{D}$ relations are studied. Starting from the ω-chains of groups with length (d, d′) and the homomorphism from the ω-chains of groups with length (d, d′) to the group, by using Bruck-Reilly expansion, the regular simple ω²-semigroups with ω_{d, d′}-type $\mathcal{D}$ relations are obtained. It is proved that any regular simple ω²-semigroup with ω_{d, d′}-type $\mathcal{D}$ relations can be constructed by Bruck-Reilly extension, and the structure theorem of this kind of semigroups is obtained.