complete_semilattices - MathStructures

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=====Complete semilattices=====

Abbreviation: **CSlat**
====Definition====
A \emph{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 
$z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |second-order |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

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

====Subclasses====
[[Complete lattices]] 

====Superclasses====
[[Directed complete partial orders]] 


====References====

[(Ln19xx>
)]

Trace: • partially_ordered_semigroups • relative_stone_algebras • finite_commutative_semigroups_with_zero • brouwerian_algebras • ordered_abelian_groups • first-order_classes • partial_semigroups • finite_semigroups_with_zero • bilinear_algebras • idempotent_semirings