Antireflexive

Antireflexive is a property used primarily in logic, mathematics, and related fields to describe a binary relation that never relates an element to itself. A relation R on a set A is antireflexive (also called irreflexive) if for every x in A, the pair (x, x) is not in R; equivalently, no element is related to itself. The term contrasts with reflexive relations, where every element is related to itself.

Common examples help illustrate the concept. The less-than relation (<) on any ordered set is antireflexive because no number is strictly less than itself. The relation "not equal to" (≠) is also irreflexive since no x satisfies x ≠ x. In contrast, the divisibility relation on positive integers is not irreflexive because every number divides itself (x divides x).

Relations and their interaction with other properties are important in theory and applications. An antireflexive and transitive relation is known as a strict partial order; if it is also connected (or total), it becomes a strict total order. In graph terms, a directed graph is loopless (has no edges from a node to itself) precisely when its adjacency relation is antireflexive.

Terminology varies slightly by author, but antireflexive and irreflexive are commonly treated as synonyms. Some discussions distinguish them in specific contexts, yet both describe the absence of self-relations. This concept is used across areas such as order theory, graph theory, and formal logic to reason about comparisons and hierarchical structures without self-reference.