Standard multiplication and addition of real or complex numbers; Group multiplication; vector addition; matrix multiplication... all these are associative.

Non-associative examples include multiplication of octonians and Lie brackets.

Associativity means that you can make sense of a*b*c without having to put in the parenthesis -- that is a*(b*c) = (a*b)*c

isn't this a bit basic? or ..what its importance?

for instance if there exists any discussion about such sets ,I don't know the reason why we learnt thems. look ,

M ≠ Ø < M , o > provides qualifications below.

1) every a,b ϵ M aob ϵ M (closed)

2) every a,b ϵ M (aob)oc = ao(boc) (associative)

3) e unit element , every a ϵ M eoa = aoe = a

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although there exist many many many subjects & titles at algebra , I do not remember we used such sets commonly.

I see you are using "Lie Algebra" in your papers

what is the usage of this definition : this is "Monoid",but has it importance? )

this set does not provide the last requirement to be group!

last requirement :

4)every a,b ϵ M one x must exist such that aox=b ˅

,4* A) e ϵ M any a ϵ M , aoe = eoa ˄ B) for any a ϵ M , a* ϵ M such that aoa* = e

4 and 4* requirements are equivalent.

and was the binary oparetaion there "o"?

correctance: e is not unit element ,it should be available only. but not element of M as I know.

**Edited by blue89, 5 September 2016 - 01:55 PM.**