$f(a)f(b)-f(c)f(d) = (f(a)-f(c))(f(b)-f(d))$

Question

Is there an operator $f$ such that $$f(a)f(b)-f(c)f(d) = (f(a)-f(c))(f(b)-f(d))$$ $$f(a)f(b)+f(c)f(d) = (f(a)+f(c))(f(b)+f(d))$$ That would be interesting to see.

Answer 1

For $$f(a)f(b) - f(c)f(d) = (f(a)-f(c))(f(b)-f(d))$$ set $a=c$ to have $$f(a)(f(b)-f(d)) = 0$$ so if the codomain is an integral domain, $f$ must be a constant.

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For $$f(a)f(b)+f(c)f(d)=(f(a)+f(c))(f(b)+f(d))$$ setting $a=b=c=d$ and cancelling common terms, we have $2f(a)^2 = 0.$ If the codomain is an integral domain with characteristic $\neq 2,$ then $f=0.$