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In Hebrew, the operation of deducing $$ab=ac \Rightarrow b=c$$ is called "צימצום" [tsimtsum] which translates roughly to "elimination" or "reduction"

More specifically for non-abelian groups the above formula would be "Right-elimination" or "Right-sided elimination", and $$ ba=ca \Rightarrow b=c$$ would be "Left elimination"

What is the accepted terminology for this in English?

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    $\begingroup$ What about simply 'cancelation'? $\endgroup$
    – user376921
    Commented Dec 20, 2019 at 10:06
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    $\begingroup$ Just saying: This deduction is possible only if you know for a fact that $a ≠ 0$ $$ba=bc$$ $$a(b-c)=0$$ $$a = 0 \ or \ b=c$$ And your question has nothing to do with "Abstract Algebra". Please remove that Tag. $\endgroup$ Commented Dec 20, 2019 at 10:09
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    $\begingroup$ Are you sure your terminology has correct chiralities? You've said we call elimination of a factor from the left right-elimination. $\endgroup$
    – J.G.
    Commented Dec 20, 2019 at 10:24
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    $\begingroup$ @RishiNandhaVanchi I'm currently studying a course called "Algebraic Structures" which is the topic of this question, but I couldn't find an "Algebraic Structures" tag, so I thought abstract-algebra would be a close enough fit. is this not the case? $\endgroup$
    – Toldry
    Commented Dec 20, 2019 at 10:54
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    $\begingroup$ The 'algebra' tag is not allowed. Please choose a more specific tag, for instance abstract-algebra or algebra-precalculus. $\endgroup$
    – Toldry
    Commented Dec 20, 2019 at 11:02

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That is called the (left or right) cancellative property.

If you're talking about groups, you can describe the operation as pre-multiplying (or post-multiplying) by $a^{-1}$. In monoids or semi-groups with the cancellative property, you cannot describe it like that because $a^{-1}$ does not formally exist, so you just call it applying the cancellative property.

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