I had an exam this morning and the first question was in a test form ,the question goes:
Imagine there is a binary operation $a*b=max(a,b)$ on N . Which option is correct ?
a)$(N,*)$ is not abelian. which is obviously wrong
b)$(N,*)$ has an identity element.
c)(N,*) has associative property. which is obviously right
d)(N,*) is a group. which is not right.
C is obviously right so this makes b wrong But why ? Isn’t 1 an identity element for this binary operation?