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.
Nice! To make this familiar to the unwashed masses, write [a,b] = ab^{-1}.
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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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Can you really derive the associative law from the 3 axioms you listed? That's [[[a,e],[b,e]], [c,e]] = [[a,e], [[b,e],[c,e]]
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Yeah, but you had me worried for a minute. First, note that we have: [c,b] = [[c,c],[b,c]] = [e,[b,c]], which I'd like to write as [c,b] = [b,c]⁻¹.
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Is it also correct to call the operator “disjunction” — i.e. the items from the two sets a & b that are not in both? Perhaps “exclusive-or” from programming is more accurate name?
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No - for one thing, "exclusive-or" is a commutative operation while [a,b] = ab^{-1} is not. Second, a and b aren't sets.
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Čini se da učitavanje traje već neko vrijeme.
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