List of First-order Theories - Lattices

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:

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.