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Contents

Definition

A completely distributive lattice is a

Properties

Algebraic lattices

Proposition

The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattices.

This appears as (Caramello, remark 4.3).

Constructive complete distributivity

A complete lattice is called constructive completely distributive if the join-assigning morphism $DA\to A$, with $DA$ the poset of downsets. This is equivalent to complete distributivity if and only if the axiom of choice holds; see (WoodFawcett). Constructive completely distributive lattices are an example of continuous algebras for a lax-idempotent 2-monad.

References

• Richard Wood? and Barry Fawcett, “Constructive complete distributivity. I”. Math. Proc. Camb. Phil. Soc. (1990), 107, 81

Revised on November 18, 2013 10:33:03 by Mike Shulman (192.195.154.58)