$\widetilde{\cal H}$-Supercryptogroups Having Regular Band Congruence.

We consider a generalized superabundant semigroup within the class of semiabundant semigroups, called a supercryptogroup since it is an analogy of a cryptogroup in the class of regular semigroups. We prove that a semigroup S is an $\widetilde{\cal H}$-regular supercryptogroup if and only if S can be expressed as a refined semilattice of completely $\widetilde{\cal J}$-simple semigroups. Some results on regular cryptogroups are extended to $\widetilde{\cal H}$-regular supercryptogroups. Some results on superabundant semigroups are also generalized. [ABSTRACT FROM AUTHOR]