Lattices
Lattices can be considered either as special sorts of partially ordered sets, with a signature consisting of one binary relation symbol ≤, or as algebraic structures with a signature consisting of two binary operations ∧ and ∨. The two approaches can be related by defining a≤ b to mean a∧b=a.
For two binary operations the axioms for a lattice are:
Commutative laws: | ||||
Associative laws: | ||||
Absorption laws: |
For one relation ≤ the axioms are:
- Axioms stating ≤ is a partial order, as above.
- (existence of c=a∧b)
- (existence of c=a∨b)
First order properties include:
- (distributive lattices)
- (modular lattices)
Completeness is not a first order property of lattice.
Read more about this topic: List Of First-order Theories