Lattice (order)
Distributivity

Since lattices come with two binary operations, it is natural to ask whether one of them distributes over the other, that is, whether one or the other of the following dual laws holds for every three elements a, b, c ∈ L, {\displaystyle a,b,c\in L,}:

A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a distributive lattice.The only non-distributive lattices with fewer than 6 elements are called M and N; they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have a sublattice isomorphic to M or N. Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).

For an overview of stronger notions of distributivity that are appropriate for complete lattices and that are used to define more special classes of lattices such as frames and completely distributive lattices, see distributivity in order theory.
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