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,