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[anti] intransitive discrete metric?

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Hi there, I haven't been on here for almost a decade but I was wondering if someone could have some ideas on a binary (ternary would can also be acceptable) , discrete metric which is intransitive, that is fully, not partly, i.e. 


for all x,y,z: xRy ^ yRz => !(xRz).


I see various ideas in topologies which are not dense and so on but it must be a discrete metric function. 

I been brainstorming for a good while but I can't quite come up with a solid solution, so ideas are very welcome.

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Hmmm, used to be quite intelligent chaps and lasses on here.. where they all gone? :/


I don't have enough time to solve this currently. Looking for free expertise from SFN! ,)

Ah .. it is clear, that we always look back and cherish the old times, since it was in a state of less entropy aye.. generally speaking. 

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I could elaborate on several paths of execution or strategy, but I think it is simplest to leave it as it is. Since no one is replying with any mathematical input, this is one for the future. 

However. Should someone wish to attempt solutions, I would a discrete relation R: x <-> y, such that with a,b,c in Z(+): if aRb and bRc, then there is no aRc.   

I think ajb would have a crack at this, usually, if he is still around. 

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I'd first ask for an example that you would consider "nontrivial".


Some observations:

1) Any such "metric" would necessarily be completely non-reflexive, that is, for all x, ~xRx. That kind of defeats many of the points of metrics. 

2) The condition you've established is entirely "false-friendly", that is, if you have a relation R that satisfies your condition, and a relation R' such that xR'y implies xRy, then R' also satisfies your condition. As such, it may be interesting to study "maximal" such relations. 

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