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So in $\mathbb{R}$ and $\mathbb{C}$ you have both associative and commutative property, but as you extend to $\mathbb{H}$ you lose the commutative property, and $\mathbb{O}$ loses the associativity. What more can you lose once you extend beyond $\mathbb{O}$? Are there some lesser known or weaker properties which holds for all extensions, or are do all such rules eventually disappear as you extend?

Thanks in advance.

[Also if someone could correctly tag this question, that would be nice]

M Turgeon
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Frank Vel
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  • I will extend $\Bbb R$ by adding a new element $\star$ with the property that $1 + \star = 2$ and $2 - \star = 17$. What property have we lost? – MJD Jan 09 '15 at 14:52
  • @MJD I don't know what this property is called, but $\star$ wouldn't have any inverse in addition. – Frank Vel Jan 09 '15 at 15:06
  • You've lost more than that; you've lost the property that if $a+b=c$ then $c-b=a$. Or maybe you've lost the property that every number has an additive inverse. Or maybe you've lost the property of associativity. Or… – MJD Jan 09 '15 at 15:33
  • This question has been asked several times before. – Lucian Jan 09 '15 at 15:38
  • @Lucian Thanks, although it seems none of those questions have an accepted answer. Where should I find an answer? – Frank Vel Jan 09 '15 at 15:45
  • An accepted answer just means that the person who asked the question, who may not know any more than you, bothered to click the check mark. The lack of an accepted answer does not mean that the answers there are not correct or that they will not satisfy you. – MJD Jan 09 '15 at 15:46
  • @MJD After having read some of the answers, they don't fully answer the questions. Well isn't this sad, finding out the question has been asked before but not answered ... – Frank Vel Jan 09 '15 at 15:56
  • @fvel If they don't fully answer your questions, you should explain why. Then maybe someone can answer whatever is left that you want to know. – M Turgeon Jan 11 '15 at 19:24

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You lose associativity, but retain power associativity: see this Wikipedia article.

M Turgeon
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