Abstract
The notion of regularity for semigroups is studied, and it is shown that an unambiguous semigroup (i.e., whose
L
and
R
orders are respectively unions of disjoint trees) can be embedded in a regular semigroup with the same subgroups and the same ideal structure (except that a zero is added to the regular semigroup).
In a previous paper [1] it was shown that any semigroup is the homomorphic image of an unambiguous semigroup with the same groups and a similar ideal structure.
Together these two papers thus prove that an arbitrary semigroup divides a regular semigroup with a similar structure.
The resulting regular semigroup is finite (resp. torsion, or bounded torsion) if the given semigroup has that property.