If a, b are elements of a monoid M and a * b = b * a. Show that (a * b) * (a * b) = (a 
* a) * (b * b).

We are tasked with proving that if \( a \) and \( b \) are elements of a monoid \( M \), and they commute, meaning \( a * b = b * a \), then \( (a * b) * (a * b) = (a * a) * (b * b) \).

### Proof:

We know the following:

- \( a \) and \( b \) are elements of a monoid \( M \), so there exists an associative binary operation \( * \) (denoted as multiplication for simplicity).
- The identity element of the monoid is denoted as \( e \), and the operation is associative: \( x * (y * z) = (x * y) * z \) for any \( x, y, z \in M \).
- It is also given that \( a \) and \( b \) commute, i.e., \( a * b = b * a \).

We need to show:

\[
(a * b) * (a * b) = (a * a) * (b * b)
\]

### Left-hand side:

We start by expanding \( (a * b) * (a * b) \) using the associativity of the monoid operation:

\[
(a * b) * (a * b) = a * (b * (a * b))
\]

Since \( a * b = b * a \) by assumption (commutativity), we can replace \( b * a \) with \( a * b \):

\[
a * (b * (a * b)) = a * ((b * a) * b) = a * ((a * b) * b)
\]

Now use the associativity of the operation again:

\[
a * ((a * b) * b) = (a * a) * (b * b)
\]

Thus, the left-hand side simplifies to \( (a * a) * (b * b) \).

### Conclusion:

Since both the left-hand side and right-hand side are equal, we have shown that:

\[
(a * b) * (a * b) = (a * a) * (b * b)
\]

The proof is complete.

---

Would you like more details, or have any questions?

### 5 Related Questions:
1. What properties must a set and operation satisfy to form a monoid?
2. How does commutativity affect the structure of a monoid?
3. Can you prove a similar identity for semigroups, where an identity element is not required?
4. How would the proof change if \( a \) and \( b \) did not commute?
5. Can you extend this result to higher powers of \( a * b \)?

### Tip:
Remember that commutativity simplifies many algebraic operations, but it's not guaranteed in all algebraic structures like monoids or groups. Always verify commutativity before using it!

If a, b are elements of a monoid M and a * b = b * a. Show that (a * b) * (a * b) = (a * a) * (b * b).

Explore the proof of the identity (a * b) * (a * b) = (a * a) * (b * b) in a commutative monoid. This problem involves key concepts from abstract algebra, including commutativity and associativity in monoids. Suitable for undergraduate mathematics students.

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