If S is a semigroup then the following statements are equivalent:[2] Even though, by definition, the rectangular bands are concrete semigroups, they have the defect that their definition is formulated not in terms of the basic binary operation in the semigroup.
The approach via the definition of nowhere commutative semigroups rectifies this defect.
Using the defining properties of a nowhere commutative semigroup, one can see that for every a in S the intersection of the Green classes Ra and La contains the unique element a.
If the Cartesian product (S / R) × (S / L) is made into a semigroup by furnishing it with the rectangular band multiplication, the map ψ becomes an isomorphism.
Other claims of equivalences follow directly from the relevant definitions.