Definition talk:Anticommutative

Just noticed this now: isn't anticommutative $$a*b=-b*a$$? wikipedia wolfram --Cynic  (talk) 04:29, 19 April 2009 (UTC)

Good question. In a general algebraic structure (in particular semigroup) not every element has an inverse. So there is no concept of $$-a*b$$, and anticommutative can only be defined as $$a*b=b*a\Longrightarrow a=b$$ or its equivalent with $$\ne$$.

Trouble is, we can't logically call such a concept "non-commutative" because that just means $$\exists a, b: a b \ne b a$$, that is, there's nothing to stop $$\exists c, d: c \ne d, c d = d c$$ for a different pair of elements in the domain.


 * (Digression: When you're talking about a single-operation structure, in particular a group, there may be conditions under which $$a \circ b \ne b \circ a \Longrightarrow a \circ b = \left({b \circ a}\right)^{-1}$$, that is, the product inverting on commutation is a logical consequence of the anticommutativity as defined. Needs to be investigated.)

In the definition $$a*b=-b*a$$, we seem to be in the context of a ring (you've got two operations, $$*$$ and (by implication) $$+$$ which is where the $$-$$ comes from as the additive inverse).

It is clear that definining anticommutativity this more specific way (i.e. in a ring) is not quite compatible with the concept in its most general terms as I've defined it here, as if $$a * b = b * a$$ then that means $$b * a = - (b * a)$$ which can be satisfied with $$b = -b$$ which does not imply $$a = b$$ so more work needs to be done to tighten this up.

The sites you cited are deplorably lax in defining the context in which the concept holds. Wolfram doesn't specify it at all (wrong link you gave BTW, I've amended it) - the reader is left to assume it's a sort of general thingy which has two operations a-bit-ish which is not too useful. The wikipedia one (despite its abstruse wordiness) is not much better, unfortunately - it's as if someone has taken a chapter of something (oh I see, Bourbaki - haven't read it myself, I only have the first volume, Set Theory) and dumped it in without paying attention to the context in which it's set.

I'll put it on the back-burner.--Matt Westwood 09:45, 19 April 2009 (UTC)

Occurs to me that an operation (the way I've defined it) can not be anticommutative in a group because then you'd have $$a * e \ne e * a$$ where $$e$$ is the identity.

So I think we'll be putting a split definition in here: one for a general algy struct / semigroup (as it is now) and one for a ring / skew field / possibly even module (which is the one found in wikipedia / wolfram.

I'll sleep on it and think about it during the day / week / month / whatever. --Matt Westwood 21:56, 19 April 2009 (UTC)