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I am trying to model a problem. I have some elements $S=\{ a,b,c \}$ and a commutative operation $\cdot:S^2 \to S^2$ on them. The operation is defined as follows:

$$ a^2 = (a,a)\\ b^2 = (b,b)\\ c^2 = (c,c)\\ ab = ba = (c,c)\\ ac = ca = (b,b)\\ bc = cb = (a,a) $$

Is there any sort of nice algebraic structure hidden in this operation? A group structure or something like that would be nice. This isn't homework or anything. I am just trying to model a problem I found.

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If all the pairs have the same value in both slots, then you might as well just define multiplication to produce that value: $a^2=bc=cb=a$, $b^2=ac=ca=b$ and $c^2=ab=ba=c$. –  aelguindy Mar 11 '12 at 20:31
    
You should do what @aelguindy suggests, so that you actually have a binary operation on $S$. You don't have an identity element, which most algebraic structures do, so you might also want to formally adjoin one. There are some structures which don't have identity though, so it depends what you're modelling. –  Matt Pressland Mar 11 '12 at 21:25
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