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. Euclid, detail from The School of Athens by Raphael. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In mathematical order theory, a semilattice is a partially ordered set (poset) within which either all binary sets have a supremum (join) or all binary sets have an infimum (meet). ...

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 () and meet (). Distributivity of these two operations is then expressed by requiring that the identity The name lattice is suggested by the form of the Hasse diagram depicting it. ...

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 In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ... In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ... In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...

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. In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...

Distributivity for semilattices

Semilattices are partially ordered sets with only one of the two lattice operations, so that we speak of meet-semilattices or join-semilattices. Given that there is only one binary operation, distributivity obviously cannot be defined in the standard way. Nevertheless, because of the interaction of the single operation with the given order, the following definition of distributivity remains possible. A meet-semilattice is distributive, if for all a, b, and x: In mathematical order theory, a semilattice is a partially ordered set (poset) within which either all binary sets have a supremum (join) or all binary sets have an infimum (meet). ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...

If a ∧ b ≤ x then there exist a' and b' such that a ≤ a' , b ≤ b' and x = a' ∧ b' .

This definition is justified by the fact that given any lattice L, the following statements are all equivalent:

• 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. Distributive join-semilattices are defined dually. This definition of distributivity allows generalizing some statements about distributive lattices to distributive semilattices. In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...

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. In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ...

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 In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. ... Pointless topology is an approach to topology which avoids the mentioning of points. ... In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ...

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 {x_j,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:

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

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: In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

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 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. 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. In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. ...

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: Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ...

• G. N. Raney, Completely distributive complete lattices, Proceedings of the American Mathematical Society, 3: 677 - 680, 1952.