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:
| Classtype | variety |
|---|---|
| Equational theory | decidable in PTIME |
| Quasiequational theory | decidable |
| First-order theory | undecidable |
| Locally finite | yes |
| Residual size | 2 |
| Congruence distributive | no |
| Congruence modular | no |
| Congruence meet-semidistributive | yes |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | yes |
| Definable principal congruences | yes |
| Equationally def. pr. cong. | no |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
$\begin{array}{lr}
f(1)= &1\quad
f(2)= &1\quad
f(3)= &1\quad
f(4)= &2\quad
f(5)= &5\quad
f(6)= &15
\end{array}$