## Complete semilattices

Abbreviation: CSlat

### Definition

A complete semilattice is a directed complete partial orders $\mathbf{P}=\langle P,\leq \rangle$ such that every nonempty subset of $P$ has a greatest lower bound: $\forall S\subseteq P\ (S\ne\emptyset\Longrightarrow \exists z\in P(z=\bigwedge S))$.

##### Morphisms

Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semilattices. A morphism from $\mathbf{P}$ to $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that preserves all nonempty meets and all directed joins:

$z=\bigwedge S\Longrightarrow f(z)=\bigwedge f[S]$ for all nonempty $S\subseteq P$ and $z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$

Example 1:

### Properties

Classtype second-order

### Finite members

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