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complete_distributive_lattices [2010/07/29 15:46] (current)
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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]]
 +)]
 +
 +