Distributivity (order theory)

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In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.

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Distributive lattices

Probably the most common type of distributivity is the one defined for lattices, where the formation of binary suprema and infima provide the total operations of join (<math>\vee<math>) and meet (<math>\wedge<math>). Distributivity of these two operations is then expressed by requiring that the identity

<math>x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)<math>

holds for all elements x, y, and z. This distributivity law defines the class of distributive lattices. Note that this requirement can be rephrased by saying that binary meets preserve binary joins. The above statement is known to be equivalent to its order dual

<math>x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)<math>

such that one of these properties suffices to define distributivity for lattices. Typical examples of distributive lattice are totally ordered sets, Boolean algebras, and Heyting algebras.

Semilattices - distributivity with one operation

Semilattices are partially ordered sets for which only one of the lattice operations is available, leading to the concepts of meet-semilattices and join-semilattices, respectively. Obviously, in the presence of only one operation, distributivity cannot be defined in a classical way. Yet, due to the interaction of the meet (resp. join) operation with the given order, one can still define a reasonable notion of distributivity. A meet-semilattice is distributive, if

if a <math>\wedge<math> bx then there are elements a' and b' such that aa' , bb' and x = a' <math>\wedge<math> b'

holds for all elements a, b, and x. Distributive join-semilattices are defined dually. This definition is justified by the fact that the following statements are equivalent for any lattice L:

  • L is distributive as a meet-semilattice
  • L is distributive as a join-semilattice
  • L is a distributive lattice.

Thus any distributive meet-semilattice in which binary joins exist is a distributive lattice. Using this insight, some statements that are usually shown for distributive lattices can be generalized to distributive semilattices.

Distributivity laws for complete lattices

For a complete lattice, arbitrary subsets have both infima and suprema and thus infinitary meet and join operations are available. Several extended notions of distributivity can thus be described. For example, finite meets may distribute over arbitrary joins, i.e.

<math>x \wedge \bigvee S = \bigvee \{ x \wedge s \mid s \in S \}<math>

may hold for all elements x and all subsets S of the lattice. Complete lattices with this property are called frames, locales or complete Heyting algebras. They arise in connection with pointless topology and Stone duality. This distributive law is not equivalent to its dual statement

<math>x \vee \bigwedge S = \bigwedge \{ x \vee s \mid s \in S \}<math>

which defines the class of dual frames.

Now one can go even further and define orders where arbitrary joins distribute over arbitrary meets. However, expressing this requires formulations that are a little more technical. Consider a doubly indexed family {xj,k | j in J, k in K(j)} of elements of a complete lattice, and let F be the set of choice functions f choosing for each index j of J some index f(j) in K(j). A complete lattice is completely distributive if for all such data the following statement holds:

<math> \bigwedge_{j\in J}\bigvee_{k\in K(j)} x_{j,k} =
        \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}
<math>

Complete distributivity is again a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Completely distributive complete lattices (also called completely distributive lattices for short) are indeed highly special structures. 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 a 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

<math> \bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\{ \bigwedge Z \mid Z\in S^\# \} <math>

The operator ( )# might be called the crosscut operator. The latter version of complete distributivity only implies the original notion when admitting the Axiom of Choice. However, the latter version is always equivalent to the statement:

<math> \bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\bigcap S<math>

for all sets S of subsets of a complete lattice.

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

  • L is completely distributive (in the original sense).
  • 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 Lop 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. The final theorem also explains why completely distributive lattice are so special. There is still more to say about complete distributivity and its intuitionistic variants: see the article on completely distributive lattices.

Literature

Distributivity is a basic concept that is treated in any textbook on lattice and order theory. See the literature given for the articles on order theory and lattice theory. More specific literature includes:

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