Finitary Relation - Transitive Relations

Transitive Relations

Transitive relations are binary relations R on a single set X where for all a, b, c in X, aRb and bRc implies aRc. Transitive relations fall into two broad classes, equivalence relations and order relations. Equivalence relations are also symmetric and reflexive, while order relations are antisymmetric (complete order) or asymmetric (partial order) and may be reflexive (inclusive order) or anti-reflexive (strict order). The algebraic structure of equivalence relations builds on transformation groups; that of order relations builds on lattice theory. For more on relations and mathematics, from a philosophical standpoint, see Lucas (1999: chpt. 9).