 Rectangular bands are also the "T"-algebras, where "T" is the monad on Set with "T"("X")="X"×"X", "T"("f")="f"×"f", formula_5 being the diagonal map formula_6, and formula_7.A normal band is a band "S" satisfying
This is the same equation used to define medial magmas, and so a normal band may also be called a medial band, and normal bands are examples of medial magmas.We can also say a normal band is a band "S" satisfying

A left-regular band is a band "S" satisfying


If we take a semigroup and define "a" ≤ "b" if and only if "ab = b", we obtain a partial ordering if and only if this semigroup is a left-regular band.