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Suppose you have a singleton reduction system $\{(x^2y,yx)\}$. Does such a system lead to a normal form on the corresponding $k$-algebra $k\langle x,y\rangle$, where $k$ is a commutative, associative unital ring?

How does one tell that it is length reducing and has no overlaps? For instance, what about a monomial $y^2x$, doesn't the reduction system send this to $yx^2y$? Maybe I'm going about this wrong, how do I know what I'm supposed to check?

Thanks,

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    It is length reducing and has no overlaps, so of course. Perhaps you are worried that the normal forms are not very pretty; the reversed rule would look better, but this does not matter for your question.2012-04-13
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    Thanks @JackSchmidt, I didn't know it was so obvious. I don't see how you saw that so quickly though. Do you mind explaining why it is length reducing and has no overlaps? Does the latter just follow since the reduction has only one reduction rule?2012-04-14
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    xxy is longer than yx, so every rule reduces the length of words it is applied to, "length reducing" and therefore noetherian (poset of rule applications). The left hand sides of rules have no overlaps, because xxy does not overlap xxy. This means that all reductions can be brought together, confluence. In particular, every word has a unique normal form given by applying the rules until they don't apply (words that are alternating products of y^i and x for i>0). You are dealing with the semigroup ring over a semigroup given by a confluent rewriting system, so we need not look at addition.2012-04-16
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    In particular, most people mean x^2y → yx, so you are applying your rule backwards. If you apply your rule backwards, it still obviously has no overlaps since yx does not overlap with yx, but noetherian is proved differently, sometimes called the wreath product or recursive ordering. In particular, the backwards rule has normal forms exactly (k-linear sums of) x^i y^j and this is a standard twisted polynomial ring.2012-04-16
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    Dear @Jack, thanks! I think I have a better feel for how this sort of thing works.2012-04-17

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