initialsegment

An initial segment of a partially ordered set P is a subset I ⊆ P with the property that if x ∈ I and y ≤ x in P, then y ∈ I. Such a subset is often called a downward-closed set or a lower set. The opposite notion is the final segment, or upper set: if x ∈ F and x ≤ y, then y ∈ F.

The family of initial segments is closed under arbitrary unions and intersections; hence they form a complete lattice under inclusion. For any subset A ⊆ P, the smallest initial segment containing A is the downward closure of A, denoted ↓A = {y ∈ P : y ≤ a for some a ∈ A}. In particular, for a single element x, ↓x = {y ∈ P : y ≤ x} is the principal initial segment determined by x. Not every initial segment is principal in general.

Examples: In the natural numbers with the usual order, the initial segments are ∅, {0}, {0,1}, {0,1,2}, ..., and the whole N. In the real numbers with ≤, the initial segments are ∅, (-∞, a) and (-∞, a] for a ∈ R, and R itself.

In ordinal theory, every ordinal α is the set of all β with β < α, i.e., a canonical initial segment of the ordinal itself. The concept also has topology connections: the down-sets of a poset P are the closed sets in the Alexandrov topology, while the up-sets are the open sets.

See also: down-set, order ideal, principal down-set, final segment.