Lattice (order) - Lattices As Posets

Lattices As Posets

A poset (L, ≤) is a lattice if it satisfies the following two axioms.

Existence of binary joins

For any two elements a and b of L, the set {a, b} has a join: (also known as the least upper bound, or the supremum).

Existence of binary meets

For any two elements a and b of L, the set {a, b} has a meet: (also known as the greatest lower bound, or the infimum).

The join and meet of a and b are denoted by and, respectively. This definition makes and binary operations. The first axiom says that L is a join-semilattice; the second says that L is a meet-semilattice. Both operations are monotone with respect to the order: a₁ ≤ a₂ and b₁ ≤ b₂ implies that a₁ b₁ ≤ a₂ b₂ and a₁b₁ ≤ a₂b₂.

It follows by an induction argument that every non-empty finite subset of a lattice has a join (supremum) and a meet (infimum). With additional assumptions, further conclusions may be possible; see Completeness (order theory) for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connections between related posets — an approach of special interest for the category theoretic approach to lattices.

A bounded lattice has a greatest (or maximum) and least (or minimum) element, denoted 1 and 0 by convention (also called top (⊤), and bottom (⊥)). Any lattice can be converted into a bounded lattice by adding a greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by (resp.) where .

A poset is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element x of a poset it is trivially true (it is a vacuous truth) that and, and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element, and the meet of the empty set is the greatest element . This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets A and B of a poset L,

and

hold. Taking B to be the empty set,

and

which is consistent with the fact that .