1

An alternative formulation of this question is: why is the set of four axioms that define a group so fundamental, rather than, for example, the same four axioms except that associativity is interchanged with commutativity?

What makes this particular set of 4 axioms so special that abstract algebra is based on it?

The four axioms for a group $(G, +)$ are:

  1. Closure: $\;\forall a, b \in G$, $\;(a+b)\in G$
  2. Associativity: For all $a, b, c \in G,\,$ $\;a+(b+c)=(a+b)+c$.
  3. Identity: $\;\exists e \in G$ such that $\forall a \in G,\; a+e=e+a = a$.
  4. Invertibility: $\forall a \in G, \exists b = -a$ such that $b+a=a+b= e$.

If we add commutativity we have a commutative group. Why not for example make commutativity the key axiom and speak of an "associative group" as a special case? Or why not something else entirely?

What's so special about this combination of 4 axioms?

Edit: For example, is there a theory of "commutative non-associative magma's with an identity and inverses"?

user56834
  • 12,925
  • I dont understand the question. These four axioms together define what is a group, they are not "special" to me in any sense. – Masacroso Feb 26 '17 at 17:58
  • I think this is a good question, but a duplicate. Re: commutativity, let me just point out that "most" objects have non-commutative (or rather, non-Abelian) symmetry groups. Indeed, Abelian groups strike me as the weird ones (even though as a non-algebraist I vastly prefer them). – Noah Schweber Feb 26 '17 at 17:59
  • @Masacroso The question is why these axioms define a class of structures which are "worth focusing on" - e.g. why are groups more compelling than loops? – Noah Schweber Feb 26 '17 at 17:59
  • @NoahSchweber oh, ok... then my answer is because groups are natural to humans from centuries more than other algebraic structures, starting with the integers. This answer is applicable to practically anything about mathematics. – Masacroso Feb 26 '17 at 18:01
  • I updated your axioms, mostly for the addiitve group you are using. – amWhy Feb 26 '17 at 18:04
  • @Masacroso I think that's disingenuous - groups are natural to humans for reasons, and we can say a bit about why groups seem to hold a more significant role in (human) mathematics than, say, loops. I agree that it's not possible to argue "groups are more fundamental than loops" objectively, but I think that's interpreting the question overly stringently. – Noah Schweber Feb 26 '17 at 18:22
  • @Noah Schweber, thanks for the answer. You say most objects have non commutative symmetry groups. Why exactly is "associativity" so important here? – user56834 Feb 26 '17 at 18:31
  • @Programmer2134 Groups tend to arise from actions, where elements of the group correspond to functions from some mathematical object to itself, and the group operation is composition. Now remember that function composition is associative . . . – Noah Schweber Feb 26 '17 at 18:55

0 Answers0