Semigroup Isomorphism/Examples/Structure with Two Operations

Examples of Semigroup Isomorphisms
Let $\struct {S_1, \circ_1, *_1}$ and $\struct {S_2, \circ_2, *_2}$ be algebraic structures such that:
 * $\struct {S_1, \circ_1}$ is isomorphic to $\struct {S_2, \circ_2}$
 * $\struct {S_1, *_1}$ is isomorphic to $\struct {S_2, *_2}$

Then it is not necessarily the case that $\struct {S_1, \circ_1, *_1}$ is isomorphic to $\struct {S_2, \circ_2, *_2}$.

Proof

 * Proof by Counterexample:

Let $\R$ denote the set of real numbers.

Let $\vee$ and $\wedge$ denote the max operation and min operation respectively.

Let $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ denote the algebraic structures formed from the above.

From Max and Min Operations on Real Numbers are Isomorphic, $\struct {\R, \vee}$ is isomorphic to $\struct {\R, \wedge}$.

Let $\struct {\R, \times}$ denote the algebraic structure formed from $\R$ under multiplication.

From Real Numbers under Multiplication form Monoid, $\struct {\R, \times}$ is a monoid and hence a semigroup.

Let $I_\R: \R \to \R$ denote the identity mapping on $\R$.

From Identity Mapping is Semigroup Automorphism we have that $\struct {\R, \times}$ is an automorphism and hence a fortiori an isomorphism.

Now consider the algebraic structures $\struct {\R, \vee, \times}$ and $\struct {\R, \wedge, \times}$.

We have from above that

We also have that:
 * $\struct {\R, \vee}$ is isomorphic to $\struct {\R, \wedge}$
 * $\struct {\R, \times}$ is isomorphic to $\struct {\R, \times}$

there exists an isomorphism $\phi$ from $\struct {\R, \vee, \times}$ to $\struct {\R, \wedge, \times}$.

Because $\phi$ is an isomorphism, it is by definition a bijection

Hence $\phi$ is both a surjection and an injection.

Then:

Then:

Thus we have that:
 * $\map \phi 0 = 0$

and:
 * $\map \phi 1 = 0$

and so $\phi$ is not an injection.

This contradicts our assertion that isomorphism.

Hence by Proof by Contradiction no such isomorphism exists.

Hence $\struct {\R, \vee, \times}$ and $\struct {\R, \wedge, \times}$ are not isomorphic.

The result follows.