Math 2.0

Order, lattices, ordered algebraic structures Definition

Distributive lattice

Definition

A lattice (L,∨,∧)(L, \vee, \wedge) is distributive if x∧(y∨z)=(x∧y)∨(x∧z)x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z) for all x,y,z∈Lx, y, z \in L.

Remarks

The condition is equivalent to its dual: x∨(y∧z)=(x∨y)∧(x∨z)x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z) for all x,y,zx, y, z. Every totally ordered set is a distributive lattice, with max⁡\max as join and min⁡\min as meet. Every Boolean algebra is a distributive lattice. The diamond lattice M3M_3 and the pentagon lattice N5N_5 are the basic lattices that are not distributive.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Distributive_lattice, Definition
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