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 (

The above statement is known to be equivalent to its order dual such that one of these properties suffices to define distributivity for lattices.

A semilattice is partially ordered set with only one of the two lattice operations, either a meet- or a join-semilattice.

Nevertheless, because of the interaction of the single operation with the given order, the following definition of distributivity remains possible.

For a complete lattice, arbitrary subsets have both infima and suprema and thus infinitary meet and join operations are available.

[4] Distributivity is a basic concept that is treated in any textbook on lattice and order theory.