Abbreviation: Slat$_0$

A \emph{semilattice with zero} is a structure $\mathbf{S}=\langle S,\cdot,0\rangle$ of type $\langle 2,0\rangle $ such that

$\langle S,\cdot\rangle$ is a semilattices

$0$ is a zero for $\cdot$: $x\cdot 0=0$

Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices with zero. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$

Example 1:

$\begin{array}{lr} f(1)= &1\quad
f(2)= &1\quad
f(3)= &2\quad
f(4)= &5\quad
f(5)= &15\quad
f(6)= &53
\end{array}$

Semilattices with identity and zero

Semilattices