MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

There are three elements: x, y, z and a relation C:

x C y, y C z, z C x, x C x, y C y, z C z.

Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" (R), i.e.

x L x = x L y = y L x = x, y L y = y L z = z L y = y, z L z = z L x = x L z = z

x R x = x R z = z R x = x, y R y = x R y = y R x = y, z R z = z R y = y R z = z.

Similar construction produces a multi-valued logic, if to use a linear order instead of the C, but this non-associative "logic" also has some applications. Yet, I failed to find any notes about that in a book about multi-valued logic. I would be glad to know, if described construction was used somewhere earlier to provide correct references in my works.

share|cite|improve this question
I don't see a question here. – Cam McLeman Apr 22 '11 at 14:10
Dear qubeat: This post, at present, is "not a real question", meaning that you've rambled a little about an idea you've had (nothing wrong with that! the best questions include some background), but never got to a question. Maybe your question is "where can I read about multi-valued logic?", but if it's only that, then it's only borderline for MathOverflow (I would expect Math.StackExchange to be a better fit). Please read , and revise this question. If it is closed (and I expect it will be), then once you revise it, you can "flag for moderator attention". – Theo Johnson-Freyd Apr 22 '11 at 14:18
Does this have anything to do with Trintercal? – Zsbán Ambrus Apr 22 '11 at 19:40
Trintercal? I do not know. – Alex 'qubeat' Apr 22 '11 at 20:13
@Theo Johnson-Freyd, thank you for the comments and suggestions, I have seen Math.StackExchange, but afraid it won't help. – Alex 'qubeat' Apr 24 '11 at 10:45

It sounds like you are describing a situation where $a$ is more true than $b$, $b$ is more true than $c$, but nevertheless $c$ is more true than $a$. I am not sure about the best starting point in looking for relevant references, but maybe Arrow's theorem on the impossibility of a perfect voting scheme, where $a$ represents "candidate $A$ should be elected".

share|cite|improve this answer
Yes, relation C itself is known very well due to rock-paper-scissors game, but I may not find anything about L and R operations (analogues of AND, OR) derived from such relation. – Alex 'qubeat' Apr 22 '11 at 15:33

Just few hours ago I found, that the construction was used in talks of J. B. Nation “How aliens do math”, and “Logic on other planets” (here). Despite of such titles, the works look quite instructive.

share|cite|improve this answer
Soon after I posted the answer, initial question was downvoted. Anyway, I think, I must have sent an answer, if I have known about one. – Alex 'qubeat' Apr 28 '11 at 12:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.