Transitive relation

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

One of the most important properties of a binary relation. A relation $R$ on a set $A$ is called transitive if, for any $a,b,c\in A$, the conditions $aRb$ and $bRc$ imply $aRc$: equivalently if the composition $R \circ R \subseteq R$. Equivalence relations and orderings are examples of transitive relations. The universal relation, $a R b$ for all $a,b \in A$, the equality relation, $a R b$ for $a=b \in A$ and the empty (nil) relation are transitive.

The intersection of transitive relations on a set is again transitive. The transitive closure $R^*$ of a relation $R$ is the smallest transitive relation containing $R$: equivalently the intersection of all transitive relations containing $R$ (there exists at least one such, the universal relation). It can be described as $a R^* b$ if there exists a finite chain $a = a_0, a_1, \ldots, a_n = b$ such that for each $i=1,\ldots,n$ we have $a_{i-1} R a_i$.

References

[a1]	R. Fraïssé, Theory of Relations, Studies in Logic and the Foundations of Mathematics, Elsevier (2011) ISBN 0080960413
[a2]	P. R. Halmos, Naive Set Theory, Springer (1960) ISBN 0-387-90092-6
[a3]	P.M. Cohn, "Universal algebra", Reidel (1981) ISBN 90-277-1213-1 MR0620952 Zbl 0461.08001