## Complete distributive lattices

Abbreviation: **CDLat**

### Definition

A ** complete distributive lattice** is a complete lattice $\mathbf{A}=\langle A,\bigvee,\bigwedge\rangle$ such that

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

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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 ** …** 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.

### Finite members

### Subclasses

[[...]] subvariety

[[...]] expansion

### Superclasses

[[...]] supervariety

[[...]] subreduct

### References

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*Title***1**, 23–45 MRreview

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