Poorset

Poorset is a term occasionally used to denote a poset, or partially ordered set, a mathematical structure that combines a set with a binary relation that defines a partial order. In a poiset, the relation ≤ is reflexive, antisymmetric, and transitive. The set P together with ≤ is denoted (P, ≤). If every pair of elements is comparable, the structure is a total order; otherwise, it remains a poset.

Formally, a poiset consists of a set P and a binary relation ≤ on P such that for all a, b, c in P: a ≤ a (reflexivity), if a ≤ b and b ≤ a then a = b (antisymmetry), and if a ≤ b and b ≤ c then a ≤ c (transitivity). The standard terminology and notation conventional in mathematics is to refer to this structure as a poset; poiset is a rarely used variant spelling in some texts.

Examples include the power set of a set X ordered by inclusion (⊆), and the natural numbers with the usual ≤ relation. These examples illustrate posets that are not necessarily totally ordered.

Key concepts associated with poisets include subposets, chains (totally ordered subsets), and antichains (sets of pairwise incomparable elements). If every pair of elements has a least upper bound and a greatest lower bound, the poset is a lattice. Monotone (order-preserving) maps between posets are functions f with a ≤ b implying f(a) ≤ f(b).

Usage notes: poiset is a nonstandard term; poset is the preferred, widely accepted term in mathematics. Poorset may appear in historical texts or software contexts, but the standard concept remains the poset.