GREEN ♯-RELATIONS AND NORMAL ${\cal H}^\sharp$-CRYPTOGROUPS

In this paper, a new set of generalized Green's relations on a semigroup S, namely the set of Green ♯-relations, are introduced. By using these new Green ♯-relations, we prove that an ${\cal H}^\sharp$-abundant semigroup is a normal ${\cal H}^\sharp$-cryptogroup if and only if it is a strong semilattice of completely ${\cal J}^\sharp$-simple cryptogroups. Our theorem simplifies a construction theorem of normal $\tilde {\cal H}$-abundant cryptographs previously described by the authors. Some properties of the good homomorphisms between the normal ${\cal H}^\sharp$-cryptogroups are also investigated.