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I read this question, Does commutativity imply Associativity?.

And then was curious if non-commutativity implies non-associativity.

For concatenation (*), this is not true.

let A = {1,2}
let B = {3,4}
let C = {5,6}

A * B = {1,2,3,4}

does not equal

B * A = {3,4,1,2}

A * (B * C) = {1,2,3,4,5,6}

and

(A * B) * C = {1,2,3,4,5,6}

However, in general, does non-commutativity imply non-associativity?

Of course it's not true for concatenation, but perhaps for other algebras?

Community
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Kevin Meredith
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    Matrix multiplication is not commutative (in general) but it is associative. – TravisJ May 12 '15 at 02:10
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    Groups are defined to be associative but are frequently not commutative. – TravisJ May 12 '15 at 02:11
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    Moreover, commutativity does not imply associativity. Consider the rock-paper-scissor magma. – Hanul Jeon May 12 '15 at 02:12
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    If non-commutativity implied non-associativity then associativity would imply commutativity. Not to argue by authority of elders, but a lot of math would seem silly if that were true – jxnh May 12 '15 at 02:18
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    To disprove an implication, all you need is one counterexample - your concatenation case is already enough to prove that non-commutativity doesn't imply non-associativity, because you've found a case that's non-commutative but not non-associative. (And canonically, any group is associative, but many many groups are non-commutative.) – Steven Stadnicki May 12 '15 at 03:03
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    Function composition is also noncommutative but associative. (Matrix multiplication can be understood as a special case of this.) – anon May 12 '15 at 03:04

3 Answers3

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The quaternions are a noncommutative (and associative) division algebra.

Commutativity and aasociativity are totally independent, even for structures as rigid as algebras, and it would be a good use of your time to stockpile some more examples.

The octonions are a noncommutative nonassociative division algebra.

The real numbers are a commutative and associative division algebra.

The cross product on triples of elements from a field of characteristic 2 form a nonassociative but commutative algebra.

Of course, moving to groups and magma makes everything even easier. Groups are all associative, but of course there are Abelian ones and nonabelian ones.

Someone mentioned my favorite nonassociative commutative magma already: the rock paper scissors magma, where rr=r, pp=p, ss=s, rs=r, rp=p, ps=s.

I don't have a favorite nonassociative noncommutative magma, but I imagjne it is very easy to haphazardly define multiplication on a small number of symbols to make one. Say, a, b, c with ab=c and ba=b, ca=c and the other products however you wish.

rschwieb
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  • The real numbers are a commutative and associative division algebra. What does this mean since (5 / 10) / 2) != 5 / (10 / 2)? – Kevin Meredith May 12 '15 at 13:52
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    @KevinMeredith For rings, the terms commutative and associative always apply to the multiplication. (Division is of course not associative, and is not what I am talking about.) – rschwieb May 12 '15 at 15:20
  • @KevinMeredith "Division algebra" refers to the fact that division is possible - the operations being focused on are still addition and multiplication. – Noah Schweber Sep 21 '15 at 06:02
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Taking A and B to be general NxN matrices we have a non-commutative algebra. One can construct a communitavie and non-associative algebra by defining op(A,B) to be (AB+BA)/2. A system that neither would be op(A,B)=(AB-BA)/2. In both cases, AB is ordinary matrix multiplication. The latter (AB-BA)/2 is sometimes called the Possonian is is used in abstract formulations of physics.

ttw
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In particular, if this were true, then associativity would imply commutativity, by contraposition. This is clearly false, as your example shows.

BenW
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