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

$\cdot $ is idempotent: $x\cdot x=x$.

Let $\mathbf{B}$ and $\mathbf{C}$ be bands. A morphism from $\mathbf{B}$ to $\mathbf{C}$ is a function $h:B\to C$ that is a homomorphism: $h(xy)=h(x)h(y)$

$\begin{array}{lr} f(1)= &1\\ f(2)= &3\\ f(3)= &10\\ f(4)= &46\\ f(5)= &251\\ f(6)= &1682\\ f(7)= &13213\\ \end{array}$

see also finite bands and http://www.research.att.com/projects/OEIS?Anum=A058112

Rectangular bands

Semilattices

Semigroups