meetinterval

Meetinterval is a term used in order theory to describe a specific substructure of a lattice defined by two elements. In a lattice L with the usual order ≤, every pair of elements a and b has a meet a ∧ b (greatest lower bound) and a join a ∨ b (least upper bound). The meetinterval of a and b, sometimes denoted I(a,b) or meetinterval(a,b), is the closed interval of all elements x in L that lie between the meet and the join:

a ∧ b ≤ x ≤ a ∨ b.

Thus I(a,b) = { x ∈ L | a ∧ b ≤ x ≤ a ∨ b }.

The concept is particularly straightforward in the lattice of subsets of a set, where the order is by inclusion. For subsets A and B of a set X, the meet is A ∩ B and the join is A ∪ B, so the meetinterval is

I(A,B) = { C ⊆ X | A ∩ B ⊆ C ⊆ A ∪ B }.

Properties and remarks:

- I(a,b) is a sublattice of L, closed under both meet and join inherited from L.

- The endpoints of the interval are a ∧ b and a ∨ b, and the interval is nonempty since a ∧ b ≤ a ≤ a ∨ b and a ∧ b ≤ b ≤ a ∨ b.

- The concept generalizes to any lattice (not just Boolean) and provides a focused view of the local structure around the pair (a,b).

Applications and usage:

- Meetintervals are used to study the local substructure of a lattice, including embeddings and sublattice properties.

- They facilitate analysis of how two elements interact within the surrounding order, particularly in distributive or modular lattices.

See also: lattice, interval, meet, join.