Distributivity (order theory)

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. 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 hold 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 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. Every finite distributive lattice is isomorphic to a lattice of sets, ordered by inclusion (Birkhoff's representation theorem). A semilattice is partially ordered set with only one of the two lattice operations, either a meet- or a join-semilattice. 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: If a ∧ b ≤ x then there exist a and b such that a ≤ a, b ≤ b' and x = a ∧ b' . Distributive join-semilattices are defined dually: a join-semilattice is distributive, if for all a, b, and x: If x ≤ a ∨ b then there exist a and b such that a ≤ a, b ≤ b and x = a ∨ b' . In either case, a' and b' need not be unique. These definitions are 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.

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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.

Semilattice

In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.

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