Abbreviation: RBand

A rectangular band is a bands $\mathbf{B}=\langle B,\cdot \rangle $ such that

$\cdot $ is rectangular: $x\cdot y\cdot x=x$.

A rectangular band is a bands $\mathbf{B}=\langle B,\cdot \rangle $ such that

$x\cdot y\cdot z=x\cdot z$.

Let $\mathbf{B}$ and $\mathbf{C}$ be rectangular bands. A morphism from $\mathbf{B}$ to $\mathbf{C}$ is a function $h:B\rightarrow C$ that is a homomorphism:

$h(xy)=h(x)h(y)$

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$

Left-zero semigroups

Right-zero semigroups

Normal bands