Lattices

A lattice is a nonempty set with two binary operations, \wedge and \vee, such that all eight axioms listed below are true. For convenience, they have been grouped into categories.

Commutative Laws

x \wedge y = y \wedge x

x \vee y = y \vee x

Associative Laws

x \wedge (y \wedge z) = (x \wedge y) \wedge z

x \vee (y \vee z) = (x \vee y) \vee z

Idempotent Laws

x \wedge x = x

x \vee x = x

Absorption Laws

x = x \wedge (x \vee x)

x = x \vee (x \wedge x)

One example of a lattice:

  • L is the set of prepositions
  • \vee is “or”
  • \wedge is “and”