complete_lattices - MathStructures

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=====Complete lattices=====

Abbreviation: **CLat**
====Definition====
A \emph{complete lattice} is a structure $\mathbf{L}=\langle L,\bigvee,\bigwedge\rangle$ such that $\bigvee,\bigwedge$ map
subsets of $L$ to elements of $L$ and

$\langle L,\vee,\wedge\rangle$ is a [[lattices|lattice]] where $x\vee y=\bigvee\{x,y\}$, $x\wedge y=\bigwedge\{x,y\}$ and

$\bigvee S$ is the least upper bound of $S$,

$\bigwedge S$ is the greatest lower bound of $S$.

==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices. 
A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a complete homomorphism: 

$h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$

====Examples====
Example 1: $\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.

====Basic results====


====Properties====
^[[Classtype]]  |Second-order |
^[[Amalgamation property]]  |Yes |
^[[Strong amalgamation property]]  |Yes |
^[[Epimorphisms are surjective]]  |Yes |

====Subclasses====
[[Algebraic lattices]] 

====Superclasses====
[[Lattices]] 


====References====

[(Ln19xx>
)]

Trace: • frames • representable_lattice-ordered_groups • stone_algebras • multiplicative_semilattices • nonassociative_relation_algebras • distributive_lattice_expansions • commutative_ordered_semigroups • second-order_classes • boolean_groups • ordered_monoids_with_zero