A group is a set with a distinguished element e and a binary operation [ , ] satisfying: ‣ [a,b] = [[a,c],[b,c]] ‣ [a,e] = a ‣ [a,a] = e for all a,b,c.
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Nice! To make this familiar to the unwashed masses, write [a,b] = ab^{-1}.
Yeah for the rest of the year, I'm only going to think about division! Goodbye, multiplication and composition.
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It should be fun to try to formulate the notion of groupoid using only division.
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Julie and I kinda did this in Section 7 of http://arxiv.org/abs/1207.3465
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It's known that groups can be actually be specified via a single axiom, right?
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I don't know... probably. Boolean logic can be described using a single operation obeying a single axiom! But these results are like circus tricks.
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Čini se da učitavanje traje već neko vrijeme.
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