ax + xy = max(ax,ay)
let a=-1, x=1, y=2, then a(x+y) = -1 max(1,2) = -2, and ax+ay = max(-1,-2) = -1, so
in this case a(x+y) != ax + ay, so this axiom is not satisfied.
If such an element existed then there is a e in R such that for all x in R e<x, but there isb) There exists an element 0 such that for any x in the proposed vector space, x + 0 = x.
no such element in R (it would have to behave like -infty, but this is not an element of R)
Trivialy true as max(x,y)=max(y,x)c) x+y = y+x