 Semilattices are exactly commutative bands; that is, they are the bands satisfying the equation

A left-zero band is a band satisfying the equation
whence its Cayley table has constant rows.Symmetrically, a right-zero band is one satisfying
so that the Cayley table has constant columns.A rectangular band is a band "S" that satisfies
or equivalently,
The second characterization clearly implies the first, and conversely the first implies "xyz" = "xy"("zxz") = ("x"("yz")"x")"z" = "xz".