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Intervals

A convex set in a poset P is a subset I of P with the property that, for any x and y in I and any z in P, if x ≤ z ≤ y, then z is also in I. This definition generalizes the definition of intervals of real numbers. When there is possible confusion with convex sets of geometry, one uses order-convex instead of "convex".

A convex sublattice of a lattice L is a sublattice of L that is also a convex set of L. Every nonempty convex sublattice can be uniquely represented as the intersection of a filter and an ideal of L.

An interval in a poset P is a subset that can be defined with interval notation:

Whenever a ≤ b does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set {1, 2, 4, 5, 8} is convex, but not an interval.

An interval I is bounded if there exist elements a, b ∈ P {\displaystyle a,b\in P} such that I ⊆. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let P = (0, 1) ∪ (1, 2) ∪ (2, 3) as a subposet of the real numbers. The subset (1, 2) is a bounded interval, but it has no infimum or supremum in P, so it cannot be written in interval notation using elements of P.

A poset is called locally finite if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product N × N {\displaystyle \mathbb {N} \times \mathbb {N} } is not locally finite, since (1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤... ≤ (2, 1).Using the interval notation, the property " a is covered by b " can be rephrased equivalently as = { a, b }. {\displaystyle =\{a,b\}.}

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.