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I'm interested in properties of relations. Things like completeness (connected, total), transitivity, euclideanness, symmetry and so on. I am interested in the logical connections between these relations. For example, symmetry implies not asymmetry. Or a reflexive, weakly connected relation is complete.

Is there a neat summary of these sorts of properties and their connections?

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Relation between relations, huh? A compilation of properties of relation classes, and then how those properties are related?

Wikipedia on Binary relations has a table near the bottom where you can compare relation classes a little. Mostly these kinds of comparisons are straightforward to prove.

Unexpectedly, an area where these properties are manipulated and interact is in the area of Modal logic, where a given axiom implies a relation among the worlds of a Kripke structure. A number of very minor derivations are of the form "S4 + X = S5, because adding the X axiom adds the symmetric property to a transitive worlds relation which implies that it is an equivalence relation" (modulo actual correct use of those properties!).

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  • $\begingroup$ What I'm looking for is a compilation of the logical connections between these concepts $\endgroup$ – Seamus Mar 31 '11 at 17:44
  • $\begingroup$ @Seamus: did you look at the table further down on that page, having columns some of the properties, and rows some relation classes? $\endgroup$ – Mitch Mar 31 '11 at 17:53
  • $\begingroup$ @Mitch yes. I'm not interested in types of relation in that sense. I'm interested in facts like "transitivity and antisymmetry imply acyclicity". This has nothing to do with equivalence classes or other types of relations. $\endgroup$ – Seamus Mar 31 '11 at 18:01
  • $\begingroup$ @Seamus: well, it sorta does have something to do with equivalence classes, since if you add some properties together, they might imply some more properties, and a relation class name is a convenient way to refer to a bundle of properties. I modified my answer to include another area to look at where you might see some inferences made. $\endgroup$ – Mitch Mar 31 '11 at 20:21
  • $\begingroup$ @Mitch fair enough. But only some collections of relational properties have names. So it seems a long way round to get at connections among relations to talk about the types of ordering they define, for example. $\endgroup$ – Seamus Mar 31 '11 at 20:25
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Is this the kind of summary you are looking for?

http://anglocatholicninjas.wordpress.com/2007/03/20/transitive-symmetric-and-reflexive-relations/

http://en.wikipedia.org/wiki/Equivalence_relation

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  • $\begingroup$ I was looking for something more comprehensive $\endgroup$ – Seamus Mar 31 '11 at 15:06

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