Lattice

A lattice is a system $K$ such that $\forall A\in K$, $A\subset A$, and if $A\subset B$ and $B\subset A$, then $A=B$, where $=$ here means ``is included in.'' Lattices offer a natural way to formalize and study the ordering of objects using a general concept known as the Poset (partially ordered set). The study of lattices is called Lattice Theory. Note that this type of lattice is an abstraction of the regular array of points known as Lattice Points.

The following inequalities hold for any lattice:

$$(x\wedge y)\vee(x\wedge z)\leq x\wedge(y\vee z)$$

$$x\vee(y\wedge z)\leq (x\vee y)\wedge(x\vee z)$$

$$(x\wedge y)\vee(y\wedge z)\vee(z\wedge x)\leq (x\vee y)\wedge(y\vee z)\wedge(z\vee x)$$

$$(x\wedge y)\vee(x\wedge z)\leq x\wedge(y\vee(x\wedge z))$$

(Grätzer 1971, p. 35). The first three are the distributive inequalities, and the last is the modular identity.

See also Distributive Lattice, Integration Lattice, Lattice Theory, Modular Lattice, Toric Variety

References

Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.