complete_distributive_lattices

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=====Complete distributive lattices=====

Abbreviation: **CDLat**

====Definition====
A \emph{complete distributive lattice} is a [[complete lattice]] $\mathbf{A}=\langle A,\bigvee,\bigwedge\rangle$ such that

$\vee$ distributes over $\wedge$:  $x\vee (y\wedge z)=(x\vee y)\wedge(x\vee z)$

Remark: 
Click on the 'Edit text of this page' link at the bottom to add some information about complete distributive lattices

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$h(x ... y)=h(x) ... h(y)$

====Definition====
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$

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

====Basic results====


====Properties====
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

^[[Classtype]]                        |second-order  |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

====Subclasses====
  [[...]] subvariety

  [[...]] expansion


====Superclasses====
  [[...]] supervariety

  [[...]] subreduct


====References====

%[(Ln19xx>
%F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

Trace: • cancellative_commutative_semigroups • brouwerian_semilattices • partially_ordered_semigroups • kleene_algebras • semiassociative_relation_algebras • distributive_lattice_expansions