Order theory

Completely distributive lattice

In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. Formally, a complete lattice L is said to be completely distributive if, for any doubly indexed family {xj,k | j in J, k in Kj} of L, we have where F is the set of choice functions f choosing for each index j of J some index f(j) in Kj. Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Without the axiom of choice, no complete lattice with more than one element can ever satisfy the above property, as one can just let xj,k equal the top element of L for all indices j and k with all of the sets Kj being nonempty but having no choice function. (Wikipedia).

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Related pages

Continuous poset | Order theory | Order embedding | Total order | Power set | Monotonic function | Unit interval | Complete lattice | Pointwise order | Completely distributive lattice | Glossary of order theory | Partially ordered set | Duality (order theory) | Free object | Choice function | Distributive lattice | Distributivity (order theory)