Abstract
By utilizing homomorphisms and -strong semilattice of semigroups, we show that the Green (*,~)-relation H*,~ is a regular band congruence on a r-ample semigroup if and only if it is a G-strong semilattice of completely J*,~-simple semigroups. The result generalizes Petrich’s result on completely regular semigroups with Green’s relation H a normal band congruence or a regular band congruence from the round of regular semigroups to the round of r-ample semigroups.
Highlights
Introduction and PreliminariesIt is well known that the usual Green’s relations on a semigroup S play an important role in the study of the structure of regular semigroups [1,2,3,4,5,6,7]
Clifford states that a semigroup is a completely regular semigroup if and only if it can be expressed as a semilattice of completely simple semigroups, where a completely regular semigroup is a semigroup whose -class contains an idempotent
Petrich both showed that a completely regular semigroup S with its Green’s relation is a normal band congruence if and only if S is a strong semilattice of complete simple semigroups [3]
Summary
It is well known that the usual Green’s relations on a semigroup S play an important role in the study of the structure of regular semigroups [1,2,3,4,5,6,7]. Petrich both showed that a completely regular semigroup S with its Green’s relation is a normal band congruence if and only if S is a strong semilattice of complete simple semigroups [3]. B. Fountain generalized the Clifford theorem by showing that an abundant semigroup is a superabundant semigroup, that is, an abundant semigroup S with every * -class of S contains an idempotent of S if and only if S is a semilattice of completely * -simple semigroups. In order to further investigate the structure of non-regular semigroups, we have to generalize the usual Green’s relations. We call a semigroup r -ample if each ∗,∼ class and each ∗,∼ -class contain an idempotent, the concept was first mentioned by Y. For some other concepts that have already appeared in the literature, we occasionally use its alternatives, though equivalent definitions
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