Subtraction "commutative" after the first element?

Mathematics Asked by Adam B. on December 5, 2020

What is the name of this property of subtraction: $a – b -c = a – c – b$? I.e. "commutative" for everything but the first element: "when the first element is fixed, the order in which we subtract the other elements from it does not matter".

I’m sorry if this is a really basic question & has been answered elsewhere, but I’ve just not been able to find anything.

2 Answers

If a binary operation is associative and satisfies $zxy=zyx$, then the semigroup is called left normal. See "left normal bands", for example. For non-associative magmas, nobody considered this. So you can call this "left normality". The right normality is defined as $xyz=yxz$ and normality is $zxyt=zyxt$ (everything for associative operations).

Answered by JCAA on December 5, 2020

In the context you've posed it, I'm not sure it has a name.

I think the proper context for this question is that of functions. In this context, we define $f_y(x) = x-y$ and your property just becomes $$f_bcirc f_c = f_c circ f_b$$ That is, it is a commutative property of those functions.

Answered by Brian Moehring on December 5, 2020

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