On the structure of regular $tilde {{cal H}}$-cryptogroups

Abstract

We introduce the concepts of Green »-relations on $tilde {{cal H}}$-abundant semigroups. By using the generalized strong semilattice of semigroups, we show that an $tilde {{cal H}}$-cryptogroup is a regular $tilde {{cal H}}$-cryptogroup if and only if it is an $tilde {{cal H}G}$-strong semilattice of completely e $tilde {{cal J} }$ -simple semigroups. This result not only extends a known result of Petrich from the class of completely regular semigroups to the class of semiabundant semigroups but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups.