Answer to Question #161395 in Abstract Algebra for k

Question

Give a proof that the associative property is a structural property. (Start with two isomorphic structures, let the operation on one be associative and show that its image must also be associative).

Answer 1

Solution:

Proof:

Let ⟨S,∗⟩ and ⟨T,♯⟩ be two arbitrary binary structures preserved by an isomorphism f:S→T. Assume ∗ is associative.

Then we have that f((a∗b)∗c)=f(a∗b)♯c for all a,b in S.

Due to associativity we have that f(a∗(b∗c))=f(a)♯f(b∗c)

Now we show that, if ⟨S,∗⟩ and ⟨T,♯⟩ are two binary structures and f:S→T is an isomorphism between them, then ⟨S,∗⟩ is associative if, and only if, ⟨T,♯⟩ is associative. By symmetry, it suffices to assume that ⟨S,∗⟩ is associative, and show that ⟨T,♯⟩ is associative.

Suppose then that x,y,z∈T; we need to show that (x♯y)♯z=x♯(y♯z). Since f is an isomorphism, it has an inverse map $f^{−1}$  that is also an isomorphism. Therefore, there exist a,b,c∈S for which x=f(a), y=f(b) and z=f(c). Then

(x♯y)♯z=(f(a)♯f(b))♯f(c)=f(a∗b)♯f(c)=f((a∗b)∗c),

since f is a homomorphism. But, since ∗ is associative, the argument (a∗b)∗c of f is equal to a∗(b∗c). Then,

f (a∗(b∗c))= f(a)♯f(b∗c)=f(a)♯(f(b)♯f(c))= x♯(y♯z)

Hence, ⟨T,♯⟩ is associative too.

Thus,  the associative property is a structural property.