Difference between revisions of "Transitive relation"
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| − | 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 $R$. 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 $a_{i-1} R a_i$. | + | 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. |
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| + | 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==== | ====References==== | ||
<table> | <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">[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">[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> | ||
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Latest revision as of 19:34, 17 November 2023
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 |
Transitive relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transitive_relation&oldid=33958