1. The problem is to understand algebraic systems focusing on semigroups, monoids, and groups, including their properties, examples, and homomorphisms. 2. **Semigroups:** A semigroup is a set $S$ with an associative binary operation $\cdot$ such that for all $a,b,c \in S$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 3. **Example of Semigroup:** The set of natural numbers $\mathbb{N}$ with addition $(+)$ is a semigroup because addition is associative. 4. **Monoids:** A monoid is a semigroup with an identity element $e$ such that for all $a \in M$, $e \cdot a = a \cdot e = a$. 5. **Example of Monoid:** The set of natural numbers $\mathbb{N}$ with addition and identity element $0$ is a monoid. 6. **Homomorphism of Semigroups and Monoids:** A function $f: S \to T$ between semigroups (or monoids) is a homomorphism if for all $a,b \in S$, $f(a \cdot b) = f(a) \cdot f(b)$ and if monoids, also $f(e_S) = e_T$ where $e_S$ and $e_T$ are identity elements. 7. **Groups:** A group is a monoid where every element has an inverse. Formally, a set $G$ with operation $\cdot$ is a group if it is associative, has identity $e$, and for every $a \in G$, there exists $a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$. 8. **Example of Group:** The set of integers $\mathbb{Z}$ with addition is a group; the identity is $0$ and the inverse of $a$ is $-a$. 9. **Subgroups:** A subset $H$ of a group $G$ is a subgroup if $H$ itself is a group under the operation of $G$. 10. **Homomorphism of Groups:** A function $f: G \to H$ between groups is a homomorphism if for all $a,b \in G$, $f(a \cdot b) = f(a) \cdot f(b)$ and $f(e_G) = e_H$. This covers the definitions, properties, examples, and homomorphisms of semigroups, monoids, and groups.