Let X be a set with a binary operation p: X x X -->X,

abbreviated to p(x,y)=xy. Suppose p satisfies

1) x(yz)=(xy)z

2) xy=yx

3) xx=x

for all x, y, z in X. Define < on X by

x<y iff xy=y.

Prove that

a) (X,<) is a partially ordered set.

b) Each pair of elements has a least upper bound. That is, given x, y in X, there is a z in X, such that x<z, y<z, and if w in X is such that x<w, y<w, then z<w.

PLEASE HELP!!!!