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

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

y*x = xy+2ay +2bx .. (2)

Since x*y = y*x, we can equate right sides (1) and (2).

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

2ax+2by = 2bx +2ay

ax+by = bx+ay.

Therefore coefficients of x's are equal or a= b.

(x*y)*z = (xy+2ax+2by)*z = xyz +2axz+2byz+2a(xy+2ax+2by) +2bz.

= xyz +2axy +2byz+2azx +4a^2x+4aby +2bz................(3)

x*(y*z) = x*(yz+2ay+2bz) = xyz +2axy+2bxz +2ax +2b(yz+2ay+2bz)

= xyz +2axy+2byz+2bzx +2ax+4aby + 4b^2..................(4)

(3)-(4) should be zero as (x*y)*z =x*(y*z) .

Therefore (2a-2b)zx +(4a^2-2a)x +(2b-4b^2)z = 0

Therefore coefficient zx should be zero: 2a-2b = 0, or a = b.

Coefficient of x should be zer: 4a^2-2a = 0 gives 2a(2a-1) = 0, So a = 0 or 2a =1, or a = 1/2.

Similarly coefficient of z should be zero: 2b-4b^2 = 0 or 2b(1-2b) gives b = 0 or b = 1/2.

Therefore a = b =1/2. Or a = b = 0 in which case x*y *y*x = xy

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**