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Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no generators and no relations. Can I construct such a thing? It makes me think that I need "anti-generators" and "anti-relations", somehow the "union" of the generators and anti-generators causes cancellation, and same goes for the relations. Which now makes me think that I need to generalise the notion of "union".

And is it possible to construct an "anti-manifold"? Where if given some manifold J and some "anti-manifold" K, then the "connected-sum" J#K will equal the empty set?

I know that these sorts of constructions are external to what groups or manifolds are. That is an "anti-group" won't be a group, because the free product of groups will always add complexity, as does the connected sum of manifolds, unless the connected sum is with an n-sphere (of appropriate dimension).

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    Why aren't "free product" and "connected sum" in quotes? They aren't any more well-defined than any of the other terms you're using.2011-02-18
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    well sure, that's true- I'll change it. I want their notion to remain as faithful as they can however to their definition in terms of defined objects. So in a sense I want to think of them as generalisations.2011-02-18
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    I'm note sure if this is what you mean. But note that there exists an epimorphism $G * H \to G$ by sending generators of $H$ to $1$. Therefore it's always true that $G*H> 1$, unless $G=H=1$. - edit - Ignore this, since apparently your H wouldn't be a group...2011-02-18
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    That was the idea, thanks @Myself. Maybe I could think of it in terms of category theory and think of my notion of the "anti-object" in more generalised way?2011-02-18
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    Ok, so anti-groups are supposed to be a different notion, somehow dual to groups or something? But then why would the free product of a group and anti-group always be a group and not an antigroup?2011-02-18
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    You might check out "Grothendieck group" and "Representation ring" as ways that people formally create negative objects. These ideas only work well if you are comfortable with G*H being isomorphic to H*G (which is true for groups, so why not also for your anti-groups). It also means that G*(-G) = 0, but G*(-H) might be some mix of both group and anti-group. You can think of it as a positron and an anti-proton hanging out. They don't cancel as far as I know.2011-02-18
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    @Jack Schmidt, those are interesting notions! The positron and anti-proton will form and anti-hydrogen atom when they cool down btw. Not sure if the metaphor still holds then.2011-02-18

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