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 {x_j,k | j in J, k in K_j} of L, we have

\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} = \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}

where F is the set of choice functions f choosing for each index j of J some index f(j) in K_j.^[1]

Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices.^[1]

Without the axiom of choice, no complete lattice with more than one element can ever satisfy the above property, as one can just let x_j,k equal the top element of L for all indices j and k with all of the sets K_j being nonempty but having no choice function.

Alternative characterizations

Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions. For any set S of sets, we define the set S^# to be the set of all subsets X of the complete lattice that have non-empty intersection with all members of S. We then can define complete distributivity via the statement

\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\{ \bigwedge Z \mid Z\in S^\# \}\end{align}

The operator ( )^# might be called the crosscut operator. This version of complete distributivity only implies the original notion when admitting the Axiom of Choice.

Properties

In addition, it is known that the following statements are equivalent for any complete lattice L:

L is completely distributive.
L can be embedded into a direct product of chains [0,1] by an order embedding that preserves arbitrary meets and joins.
Both L and its dual order L^op are continuous posets.

Direct products of [0,1], i.e. sets of all functions from some set X to [0,1] ordered pointwise, are also called cubes.

Free completely distributive lattices

Every poset C can be completed in a completely distributive lattice.

A completely distributive lattice L is called the free completely distributive lattice over a poset C if and only if there is an order embedding $\phi:C\rightarrow L$ such that for every completely distributive lattice M and monotonic function $f:C\rightarrow M$ , there is a unique complete homomorphism $f^*_\phi:L\rightarrow M$ satisfying $f=f^*_\phi\circ\phi$ . For every poset C, the free completely distributive lattice over a poset C exists and is unique up to isomorphism.^[2]

This is an instance of the concept of free object. Since a set X can be considered as a poset with the discrete order, the above result guarantees the existence of the free completely distributive lattice over the set X.

Examples

The unit interval [0,1], ordered in the natural way, is a completely distributive lattice.^[3]
- More generally, any complete chain is a completely distributive lattice.^[4]
The power set lattice $(\mathcal{P}(X),\subseteq)$ for any set X is a completely distributive lattice.^[1]
For every poset C, there is a free completely distributive lattice over C.^[2] See the section on Free completely distributive lattices above.

References

1 2 3 B. A. Davey and H. A. Priestey, Introduction to Lattices and Order 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4
1 2 Joseph M. Morris, Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy, Mathematics of Program Construction, LNCS 3125, 274-288, 2004
↑ G. N. Raney, Completely distributive complete lattices, Proceedings of the American Mathematical Society, 3: 677 - 680, 1952.
↑ Alan Hopenwasser, Complete Distributivity, Proceedings of Symposia in Pure Mathematics, 51(1), 285 - 305, 1990.

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