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!).
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