Question

what is isomorphism?

Answer 1

Answer on Question #50441 - Math - Abstract Algebra

Let $(X, f_1, \ldots, f_n)$ and $(Y, g_1, \ldots, g_n)$ are two algebraic structures, it means that $f_i, g_i$ are maps from $X^{n_i}$ to $X$ and from $Y^{n_i}$ to $Y$ respectively, where $n_i$ is an arity of $f_i, g_i$.

Then the map $f: X \to Y$ is called an isomorphism if for every $i \leq n$ and for every $x_1, \ldots, x_{m_i} \in X$

$$f(f_i(x_1, \ldots, x_{m_i})) = g_i(f(x_1), \ldots, f(x_{m_i}))$$

For example rings have two operations $+, \times$, so the map $f: X \to Y$ is an isomorphism between two rings $(X, +, \times)$ and $(Y, +, \times)$ if for every $x_1, x_2 \in X$:

$$f(x_1 + x_2) = f(x_1) + f(x_2) \text{ and } f(x_1 \times x_2) = f(x_1) \times f(x_2).$$

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