A magma is an ordered pair $(M, \cdot)$ where $M$ is a non-empty set and $\cdot: M \times M \to M$ is a binary operation.

• The set of integers $\mathbb{Z}$ with the operation of addition $+$ is a magma.
• The set of integers $\mathbb{Z}$ with the operation of multiplication $\times$ is a magma.
• The set of integers $\mathbb{Z}$ with the operation of division $\div$ is not a magma, because division is not a well-defined operation on the set of integers.

A semigroup is an ordered pair $(S, \cdot)$ where $S$ is a non-empty set and $\cdot: S \times S \to S$ is a binary operation satisfying:

• Associativity: $\forall a, b, c \in S, (a \cdot b) \cdot c = a \cdot (b \cdot c)$

• The set of integers $\mathbb{Z}$ with the operation of addition $+$ is a semigroup.
• The set of integers $\mathbb{Z}$ with the operation of multiplication $\times$ is a semigroup.

A monoid is an ordered triple $(M, \cdot, e)$ where $M$ is a non-empty set, $\cdot: M \times M \to M$ is a binary operation, and $e \in M$ is a distinguished element satisfying:

• Associativity: $\forall a, b, c \in M, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Identity element: $\forall a \in M, e \cdot a = a \cdot e = a$

• The set of integers $\mathbb{Z}$ with the operation of addition $+$ and identity element $0$ is a monoid.
• The set of integers $\mathbb{Z}$ with the operation of multiplication $\times$ and identity element $1$ is a monoid.

A commutative monoid is an ordered triple $(M, \cdot, e)$ where $M$ is a non-empty set, $\cdot: M \times M \to M$ is a binary operation, and $e \in M$ is a distinguished element satisfying:

• Associativity: $\forall a, b, c \in M, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Identity element: $\forall a \in M, e \cdot a = a \cdot e = a$
• Commutativity: $\forall a, b \in M, a \cdot b = b \cdot a$

• The set of integers $\mathbb{Z}$ with the operation of addition $+$ and identity element $0$ is a commutative monoid.
• The set of integers $\mathbb{Z}$ with the operation of multiplication $\times$ and identity element $1$ is a commutative monoid.

A group is an ordered triple $(G, \cdot, e)$ where $G$ is a non-empty set, $\cdot: G \times G \to G$ is a binary operation, and $e \in G$ is a distinguished element satisfying:

• Associativity: $\forall a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Identity element: $\forall a \in G, e \cdot a = a \cdot e = a$
• Inverse elements: $\forall a \in G, \exists a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$

• The set of integers $\mathbb{Z}$ with the operation of addition $+$, identity element $0$, and inverse $-n$ for any integer $n$ is a group.
• The set of integers $\mathbb{Z}$ with the operation of multiplication $\times$ is not a group, because not every element has a multiplicative inverse in $\mathbb{Z}$.

An abelian group is an ordered triple $(G, \cdot, e)$ where $G$ is a non-empty set, $\cdot: G \times G \to G$ is a binary operation, and $e \in G$ is a distinguished element satisfying:

• Associativity: $\forall a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Identity element: $\forall a \in G, e \cdot a = a \cdot e = a$
• Inverse elements: $\forall a \in G, \exists a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$
• Commutativity: $\forall a, b \in G, a \cdot b = b \cdot a$

• The set of integers $\mathbb{Z}$ with the operation of addition $+$, identity element $0$, and inverse $-n$ for any integer $n$ is an abelian group.
• The set of non-zero rational numbers $\mathbb{Q} \setminus \{0\}$ with multiplication $\times$ is an abelian group.

A ring is an ordered quintuple $(R, +, \cdot, 0, 1)$ where $R$ is a non-empty set, $+: R \times R \to R$ and $\cdot: R \times R \to R$ are binary operations, and $0, 1 \in R$ are distinguished elements satisfying:

• $(R, +, 0)$ is an abelian group
• Associativity of multiplication: $\forall a, b, c \in R, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Multiplicative identity: $\forall a \in R, 1 \cdot a = a \cdot 1 = a$
• Left distributivity: $\forall a, b, c \in R, a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
• Right distributivity: $\forall a, b, c \in R, (a + b) \cdot c = (a \cdot c) + (b \cdot c)$

• The set of integers $\mathbb{Z}$ with addition $+$, multiplication $\times$, additive identity $0$, and multiplicative identity $1$ is a ring.

A field is an ordered quintuple $(F, +, \cdot, 0, 1)$ where $F$ is a non-empty set, $+: F \times F \to F$ and $\cdot: F \times F \to F$ are binary operations, and $0, 1 \in F$ are distinguished elements satisfying:

• $(F, +, 0)$ is an abelian group
• $(F \setminus \{0\}, \cdot, 1)$ is an abelian group
• Left distributivity: $\forall a, b, c \in F, a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
• Right distributivity: $\forall a, b, c \in F, (a + b) \cdot c = (a \cdot c) + (b \cdot c)$

• The set of rational numbers $\mathbb{Q}$ with addition $+$, multiplication $\times$, additive identity $0$, and multiplicative identity $1$ is a field.