It is a basic question, none the less I cannot find an answer:
A monoid is associative (with an identity)
(m1∙m2)∙m3=m1∙(m2∙m3). e∙m=m∙e=m
If you consider a monoid over natural numbers (N,+,0) for givens m1,m2,m3 you will have the same result whatever order you put the 3 elements. 1+2+3=3+2+1=5
But if you take a free monoid of Lists with Concatenation (List,++,[])
[a,b,c]++[d,e,f] does not equals [d,e,f]++[a,b,c]
So we can say the monoid (N,+,0) has extra property than (List,++,[]) or I'm I lost? What is it called?
Regards, Jean
A monoid where $a*b=b*a$ for all $a$ and $b$ is called commutative.
Commutativity together with associativity implies that you can rearrange arbitarily long strings of the operation arbitrarily -- for example:
$$ a*b*c*d*e = e*d*b*a*c $$