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Monoid of natural numbers with addition have such property, that for any $n,m, k \in \mathbb{N}$ if $n+k=m+k$ then $n=m$. Does this property have some name in English?

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I usually see it being called the cancellation property. (because what you are doing above is cancelling the $k$s in both sides).

More precisely, as has been pointed out in the comment below, this is the right cancellation property.

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    $\begingroup$ Right cancellation, to be precise. $\endgroup$ – Wojowu Jan 5 '15 at 20:56
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    $\begingroup$ Or "right cancellability" $\endgroup$ – GFauxPas Jan 5 '15 at 20:59
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This is the cancellation property. Bear in mind that it does not hold for a general monoid. The cancellative property is guaranteed by the existence of inverses, therefore, all monoids with inverses (groups) are cancellation structures.

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  • $\begingroup$ You must mean that all monoids with inverses are cancellation structures. This is because for $mk = nk$ to imply $m = n$, you need to write $(mk)k^{-1} = m(k k^{-1})$, i.e. you need associativity. For an explicit magma counterexample, take $\{e, x, y\}$ with identity $e$, $x^2 = y^2 = xy = yx = e$. In this example $x$ has inverse $x$ and $y$ has inverse $y$, and $xy = yy$, but $x \ne y$. $\endgroup$ – 6005 Jan 5 '15 at 21:30
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    $\begingroup$ @Goos, no, I didn't mean that, I was just wrong :P. Yes, we also need associativity. I'll update. $\endgroup$ – Jonathan Hebert Jan 5 '15 at 21:45
  • $\begingroup$ @Goos but in your example we have $x=xe=x(xy)=x^2y=y$ $\endgroup$ – Michal Seweryn Jan 5 '15 at 21:54
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    $\begingroup$ @Michael Goos is giving a counter-example of a general magma with inverses being cancellative, you are assuming associativity. $\endgroup$ – Jonathan Hebert Jan 5 '15 at 21:56
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    $\begingroup$ @MichalSeweryn $x(xy) \ne (x^2) y$. Magma's don't need to be associative. $\endgroup$ – 6005 Jan 5 '15 at 21:56

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