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
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.