hi can you show me the 5 semigroups of order 2 please

In semigroup theory, a **semigroup** is a set equipped with an associative binary operation. For semigroups of order 2, the set consists of two elements, typically denoted as \( S = \{a, b\} \), and we define a binary operation (like multiplication) on these elements such that the operation is closed and associative.

There are exactly 5 possible semigroups of order 2. Here are the Cayley tables (operation tables) for each:

1. **Semigroup 1 (Both elements act like a left zero element):**

\[
\begin{array}{c|cc}
\cdot & a & b \\
\hline
a & a & a \\
b & b & b \\
\end{array}
\]

In this semigroup, both elements act as identity-like elements from the left, meaning \( a \cdot a = a \), \( a \cdot b = a \), and similarly for \( b \).

---

2. **Semigroup 2 (Element \( a \) is an identity, and \( b \) is idempotent):**

\[
\begin{array}{c|cc}
\cdot & a & b \\
\hline
a & a & b \\
b & b & b \\
\end{array}
\]

In this semigroup, \( a \) is an identity-like element (it doesn't change anything when multiplied by any element), and \( b \) is idempotent (\( b \cdot b = b \)).

---

3. **Semigroup 3 (Both elements act like a right zero element):**

\[
\begin{array}{c|cc}
\cdot & a & b \\
\hline
a & b & b \\
b & b & b \\
\end{array}
\]

Here, the operation on any two elements results in \( b \), which behaves as a "right absorbing" element for any multiplication.

---

4. **Semigroup 4 (Both elements act like an identity element for each other):**

\[
\begin{array}{c|cc}
\cdot & a & b \\
\hline
a & a & a \\
b & a & b \\
\end{array}
\]

In this semigroup, \( a \cdot a = a \), and multiplying by \( b \) returns either \( a \) or \( b \), making this structure slightly less straightforward.

---

5. **Semigroup 5 (Trivial semigroup, both elements combine in a fixed way):**

\[
\begin{array}{c|cc}
\cdot & a & b \\
\hline
a & a & a \\
b & a & a \\
\end{array}
\]

This is the semigroup where every product results in \( a \), no matter which elements are multiplied together.

---

Each of these tables describes how the operation behaves on the set \( S = \{a, b\} \) under the conditions of closure and associativity.

Would you like more details about any of these semigroups?

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### 5 Related Questions:
1. What are the properties of each semigroup of order 2?
2. How do semigroups differ from groups and monoids?
3. Can you explain the importance of idempotent elements in semigroups?
4. What are the common real-world applications of semigroups?
5. How does associativity work in semigroup operations?

### Tip:
When analyzing semigroups, always check for **associativity** carefully, as it's the defining feature that distinguishes semigroups from other algebraic structures.

Explore the 5 semigroups of order 2 with detailed Cayley tables and their properties. Learn about semigroup theory, binary operations, and the concept of associativity in algebraic structures. Suitable for undergraduate mathematics students.

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