Abstract algebra studies algebraic structures.

Algebraic operation $\omega: X^n \to X$ is a mapping from the $n$-th Cartesian product $X^n$ of a set to the set $X$ itself, where $n$ is the arity of the algebraic operation: nullary operation, $n = 0$; unary operation, $n = 1$; binary operation, $n = 2$; finitary operation, $n \in \mathbb{N}$; infinitary operation, $n = \aleph_\alpha, \alpha \in \mathbb{N}$.

Algebraic system $(X, (\omega_1, \cdots), (R_1, \cdots))$ is a set $X$ with a class $(\omega_1, \cdots)$ of finitary operations and a class $(R_1, \cdots)$ of relations; such operations and relations are called basic or primitive to the algebraic system. Universal algebra or algebra is an algebraic system that has no basic relation. Relational system or model (in logic) is an algebraic system that has no basic operation.

## Group

Magma (原群), semigroup (半群), monoid (幺半群), group (群), and abelian group (交换群) are algebras $(X, ∗)$ where $∗$ is a binary operation satisfying a cumulative list of properties, defined in the following table.

Table: Group-like Algebras $(X, ∗)$ by Cumulative Properties of the Operation

Property Property Definition Algebra
closure $\forall a,b \in X, a ∗ b \in X$ magma
associativity $\forall a,b,c \in X, (a ∗ b) ∗ c = a ∗ (b ∗ c)$ semigroup
identity $\exists e \in X, \forall a \in X, e ∗ a = a ∗ e = a$ monoid
inverse $\forall a \in X, \exists b \in X, b ∗ a = a ∗ b = e$ group
commutativity $\forall a,b \in X, a ∗ b = b ∗ a$ abelian group

Examples of groups:

• Integers $(\mathbb{Z},+)$, where the identity is 0 and the inverse of $a$ is $-a$.
• Dihedral groups have underlying sets consisting of symmetries like rotations and flections, and composition as group operation.

## Ring

Ring $(X, (+, ×))$ is a set $X$ with two binary operations called addition $+$ and multiplication $×$, such that:

1. $(X, +)$ is an abelian group;
2. $(X, ×)$ is a monoid;
3. Multiplication left and right distributes over addition: $a × (b + c) = (a × b) + (a × c)$, $(a + b) × c = (a × c) + (b × c)$;

For a ring $(X, (+, ×))$, zero $0$ denotes the identity of $+$ (additive identity), and $1$ denotes the identity of $×$ (multiplicative identity). Subtraction $-$ is the inverse operation of addition $+$.

Basic properties:

• Multiplication by $0$ annihilates $X$: $0 × a = a × 0 = 0$;
• $-1 × a = -a$;
• The zero ring or trivial ring: if $0=1$ in a ring, then it is the only element of the ring.
• The additive identity $0$ and the additive inverse $-a$ of each element $-a$ are unique.

Examples of rings:

• Integers $(\mathbb{Z}, (+,×))$, where the additive identity is $0$, the additive inverse of $a$ is $-a$, and the multiplicative identity is $1$.
• Modular arithmetic $\mathbb{Z}/n\mathbb{Z}$
• $\mathcal{M}_n(R)$, where the underlying set is all n-by-n matrices over an arbitrary ring $R$, with matrix addition and matrix multiplication as corresponding operations. It is a special case of matrix ring.
• Polynomial ring over $R$, $R[t]$, consists of the set of polynomials in one or more variables $t$ with coefficients in another ring $R$, often a field.

Semiring is an algebra similar to a ring, but without additive inverses. Commutative ring is a ring whose multiplication is commutative. Division ring is a ring with multiplicative inverses for all nonzero elements; division is the inverse operation of multiplication $+$, defined for all nonzero elements of a division ring.

### Field

Field $(X, (+, ×))$ is a commutative division ring.

Examples of fields: finite field with four elements; the fields of rational numbers $(\mathbb{Q}, (+,×))$, real numbers $(\mathbb{R}, (+,×))$, and complex numbers $(\mathbb{C}, (+,×))$.

Unary algebra.

Banach algebra.