1. **Problem Statement:** Understand algebraic systems focusing on semigroups, monoids, and groups, including their properties, examples, and homomorphisms. 2. **Algebraic Systems:** A set with one or more operations defined on it. 3. **Semigroup:** A set $S$ with an associative binary operation $\cdot$ such that for all $a,b,c \in S$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 4. **Example of Semigroup:** $(\mathbb{N}, +)$, natural numbers with addition. 5. **Monoid:** A semigroup with an identity element $e$ such that for all $a \in M$, $e \cdot a = a \cdot e = a$. 6. **Example of Monoid:** $(\mathbb{N}_0, +)$ natural numbers including zero with addition; zero is the identity. 7. **Homomorphism of Semigroups/Monoids:** A function $f: S \to T$ between semigroups/monoids preserving the operation: $f(a \cdot b) = f(a) \cdot f(b)$ and for monoids $f(e_S) = e_T$. 8. **Groups:** A monoid where every element has an inverse $a^{-1}$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$. 9. **Example of Group:** $(\mathbb{Z}, +)$ integers with addition; inverse is the negative number. 10. **Subgroups:** A subset $H$ of a group $G$ that is itself a group under the operation of $G$. 11. **Homomorphism of Groups:** A function $f: G \to H$ preserving group operation and identity, and inverses: $f(a \cdot b) = f(a) \cdot f(b)$, $f(e_G) = e_H$, and $f(a^{-1}) = (f(a))^{-1}$. This covers definitions, properties, and examples of semigroups, monoids, groups, subgroups, and their homomorphisms.