Given the law of composition x*y=xy+2ax+2y determine a and b if the law is communicative and associative.

If the law is commutative
then:

x*y = y*x

We'll write
the law of composition for x*y:

x*y = xy+2ax+2y
(1)

y*x = yx + 2ay + 2x
(2)

We'll put (1) =
(2):

xy+2ax+2y = yx + 2ay +
2x

We'll eliminate like
terms:

2ax+2y = 2ay + 2x

The
coefficients of x from both sides have to be equal:

2a =
2

We'll divide by
2:

a =
1

If the law is associative
then:

 (x*y)*z = x*(y*z)
xy+2ax+2y

(xy+2ax+2y)*z =
x*(yz+2ay+2z)

(xy+2ax+2y)z + 2a(xy+2ax+2y) + 2z =
x(yz+2ay+2z) + 2ax + 2(yz+2ay+2z)

We'll remove the
brackets:

xyz + 2axz + 2yz + 2axy + 4a^2x + 4ay + 2z = xyz
+ 2axy + 2xz + 2ax + 2yz + 4ay + 4z

We'll eliminate like
terms (the bolded one):

xyz +
2axz + 2yz + 2axy + 4a^2x +
4ay + 2z = xyz +
2axy + 2xz + 2ax + 2yz +
4ay + 4z

We'll factorize by
2ax to the left side and by 2z to the right side:

2ax(z+2a)
+ 2z = 2z(x+2) + 2ax

There is no b in the
expression!

The law is commutative and not
"communicative"!