Connected relation

A binary relation is connected if, for every two distinct elements in its domain, at least one is related to the other. It need not be reflexive or transitive.

Connect

Binary relation: A relation between ordered pairs of elements, usually represented as a set of pairs. Connectedness is a condition imposed on a binary relation.
Strict total order: A strict partial order in which every pair of distinct elements is comparable. This is the standard order structure obtained by adding connectedness to a strict partial order.
Reflexive relation: A relation in which every element is related to itself. Connectedness only constrains distinct pairs and says nothing about self-relatedness.
Rational preference: A preference relation satisfying consistency conditions such as completeness and transitivity. Connectedness supplies completeness, ensuring every pair of options can be ranked.
Comparable elements: Two elements are comparable under a relation when at least one is related to the other. Every distinct pair in a connected relation must be comparable.
Total preorder: A reflexive, transitive, connected relation that may treat distinct elements as tied. It combines connected comparison with reflexivity and transitivity.
Transitive relation: A relation where relating the first element to the second and the second to the third implies relating the first to the third. Connectedness compares pairs but does not impose this chain condition.
Utility function: A function assigning numerical values that represent an ordering or preference relation. A connected preference relation can often be represented by a utility ranking under suitable assumptions.
Domain of a relation: The set of elements that occur as first components of a relation's ordered pairs. The condition applies to distinct elements in the relation's specified domain.
Partial order: A reflexive, antisymmetric, transitive relation that may leave some elements incomparable. Partial orders permit incomparable pairs, whereas connected relations do not.