So, I was looking into fields and was studying up on them, and noticed that associativity seems to be a rephrased version of commutavity.

 

For a set [latex]S[/latex], with operations [latex]*[/latex] and [latex]o[/latex], and given that the set is commutative, but not necessarily associative, we can rephrase the question of associativity like so:

 

[latex] a * (b * c) = (a * b) * c \implies (b * c) * a = (a * b) * c \implies (b * c) * a = (b * a) * c [/latex]

 

This is rather obvious, especially if it is associative, but we can still phrase it differently:

 

For a given [latex] b [/latex], where [latex] x|y = (b * x) * y [/latex], is [latex] | [/latex] commutative.

 

In other words, associativity implies commutativity, but commutativity does not imply associativity.

 

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Is my reasoning erroneous? Is there something I am missing?

never mind. Matrix multiplication... FFS. How did I forget that. I even double checked that myself the other day.

Edited March 27, 20178 yr by Mathematical

Do you have a question?

Not anymore, I was trying to put into words what I was thinking, and had gotten it in my head associativity implied commutativity and visa versa. I was trying to get my reasoning in a form I could more easily criticize, and after I had done so, I had remembered matrices.

 

As such, I simply edited the post, and made it clear that I was wrong, and it should have been clear there was no need for any further discussion. However, that seems to not be the case, so let me be clear: I was wrong, and there is no further need for discussion.