What are a and b if x*y = y*x and (x*y)*z= x*(y*z) x*y = xy + 2ax + 2by

If the law has the property x*y = y*x, then the law is
commutative.

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

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

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

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

xy+2ax+2by = yx + 2ay +
2bx

We'll eliminate like
terms:

2ax+2by = 2ay + 2bx

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

2a =
2b

We'll divide by
2:

a =
b

If the law has the property  (x*y)*z =
x*(y*z), the law is associative:

 (x*y)*z =
x*(y*z)

(xy+2ax+2by)*z = x*(yz+2ay+2bz)
(3)

But a = b and we'll re-write
(3):

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

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

We'll remove the
brackets:

xyz + 2axz + 2ayz + 2axy + 4a^2x + 4a^2y + 2z =
xyz + 2axy + 2axz + 2ax + 2yz + 4ay + 4az

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

xyz +
2axz + 2ayz + 2axy + 4a^2x +
4a^2y + 2z = xyz + 2axy +
2axz + 2ax + 2yz + 4ay +
4az

Since the law is associative, the correspondent
coefficients from both sides:

2a =
2

We'll divide by
2:

a =
1

Since a = b, b  =1,
too.

The law of composition is
determined and it's expression
is:

x*y =
xy+2x+2y