View source for Transitive relation

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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 relation]]s and [[Order (on a set)|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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) {{ISBN|0080960413}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. R. Halmos, ''Naive Set Theory'', Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  P.M. Cohn, "Universal algebra", Reidel (1981) {{ISBN|90-277-1213-1}} {{MR|0620952}} {{ZBL|0461.08001}}</TD></TR>
</table>