Croisot gave a definition
of (m,n)-regularity which he then showed defined four logically distinct
classes of semi-groups. However, semigroups with nilpotent elements did
not fall within his classification. Our generalization of (m,n)0-regularity
remedies this exclusion; countably many distinct classes of semi-groups are
defined.
In particular we investigate the structure of semigroups which are (2,2)0-regular.
We show that a semigroup S is in this class precisely when for each x ∈ S either
x2= 0 or x2∈ Hx. Further, each regular 𝒟-class together with 0 of such a semigroup
is itself a completely 0-simple semigroup. The (2,2)0-regularity condition is
specialized to that of absorbency: for each a,b ∈ S either ab = 0 or ab ∈ (Ra∩ Lb).
We show that a regular absorbent semigroup is just a mutually annihilating
collection of completely 0-simple semigroups.
