1
$\begingroup$

This question already has an answer here:

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]

$\endgroup$

marked as duplicate by MJD, Venus, Ahaan S. Rungta, Davide Giraudo, Newb Jan 9 '15 at 18:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ 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? $\endgroup$ – MJD Jan 9 '15 at 14:52
  • $\begingroup$ @MJD I don't know what this property is called, but $\star$ wouldn't have any inverse in addition. $\endgroup$ – Frank Vel Jan 9 '15 at 15:06
  • $\begingroup$ 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… $\endgroup$ – MJD Jan 9 '15 at 15:33
  • $\begingroup$ This question has been asked several times before. $\endgroup$ – Lucian Jan 9 '15 at 15:38
  • $\begingroup$ @Lucian Thanks, although it seems none of those questions have an accepted answer. Where should I find an answer? $\endgroup$ – Frank Vel Jan 9 '15 at 15:45
1
$\begingroup$

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

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.