Abbreviation: Slat$_1$

A \emph{semilattice with identity} is a structure $\mathbf{S}=\langle S,\cdot,1\rangle$ of type $\langle 2,0\rangle$ such that

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

$1$ is an indentity for $\cdot$: $x\cdot 1=x$

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

$h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

Example 1:

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

Semilattices with identity and zero

Semilattices