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

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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: $h(x \ldots y)=h(x) \ldots h(y)$

Definition

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

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

Subclasses

[[...]] subvariety
[[...]] expansion

Superclasses

[[...]] supervariety
[[...]] subreduct

References

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1) %F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview

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