Algebra

(Abstract) algebra studies algebraic structures.

Algebraic operation $\omega: X^n \to X$ is a mapping from the a Cartesian power of a set to the set itself, where $n$ is the arity (元数) 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 structure $((\omega_i)_{i \in I}, (R_j)_{j \in J})$ on a set is a class of finitary operations and a class of finitary relations on the set. Algebraic system $(X, (\omega_i)_{i \in I}, (R_j)_{j \in J})$ is a set endowed with an algebraic structure. Basic or primitive operations and relations of an algebraic system are those in its algebraic structure. Universal algebra (泛代数) or algebra is an algebraic system with no basic relation. Relational system or model (in logic; 模型) is an algebraic system with 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.

Left action (作用) $g \cdot x$ of a group $(G, ∗)$ on a set $X$ is a map $G \times X \mapsto X$ that satisfies identity and associativity: $1 ∗ x = x$; $g_1 \cdot (g_2 \cdot x) = (g_1 ∗ g_2) \cdot x$. Right action (右作用) of a group on a set is similarly defined, only with group elements appearing on the right.

Ring

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

$(X, +)$ is an abelian group;
$(X, ×)$ is a monoid;
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 (半环) is an algebra similar to a ring, but without additive inverses.

Commutative ring (交换环) is a ring whose multiplication is commutative. Quadratic form $q: \mathcal{R}^n \mapsto \mathcal{R}$ over a commutative ring $\mathcal{R}$ is a homogeneous quadratic polynomial with coefficients from the ring: $q(x) = \sum_{i=1}^n \sum_{j=i}^n q_{ij} x_i x_j$ where $x_i, q_{ij} \in \mathcal{R}$. Kronecker matrix $A$ of a quadratic form is the symmetric matrix defined by $a_{ij} = q_{ij} + q_{ji}$.

Field

Division ring (除环) 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 (域) $(X, (+, ×))$ 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 $(\mathbb{C}, (+,×))$.

Module

Module (模) $(V, (+, \cdot_{\mathcal{R}}))$ over a commutative ring $(\mathcal{R}, (+, ×))$ is a set $V$ endowed with a binary operation $+: V^2 \mapsto V$ and a map $\cdot_{\mathcal{R}}: \mathcal{R} \times V \mapsto V$. Module is a generalization of vector space. Submodule of an $\mathcal{R}$-module is an $\mathcal{R}$-module consisting of a subset and the same module structure. Direct product (直积) $(\prod_{\alpha \in A} V_\alpha, (+, \cdot))$ of an indexed family $(V_\alpha)_{\alpha \in A}$ of $\mathcal{R}$-modules is the $\mathcal{R}$-module consisting of the Cartesian product and addition and scalar multiplication defined as: $(v_\alpha) + (v'\alpha) = (v\alpha + v'\alpha)$, $c (v\alpha) = (c v_\alpha)$, where $(v_\alpha)$ is an "A-tuple". External direct sum (直和), or coproduct (余积) in the category of modules, $(\oplus_{\alpha \in A} V_\alpha, (+, \cdot))$ of an indexed family of $\mathcal{R}$-modules is the submodule of their direct product that consists of A-tuples with a finite number of nonzero components. The direct product and the direct sum of a finite family of modules are identical. We say an $\mathcal{R}$-module $W$ is the internal direct sum of an indexed family $(V_\alpha)_{\alpha \in A}$ of its submodules if every element of the module has a unique expression as a finite sum $\sum_\alpha v_\alpha$ of elements of the submodules: $W = \oplus_{\alpha \in A} V_\alpha$.

Misc

Algebra $(A, (+, \cdot_{\mathbb{R}}, ×))$ over $\mathbb{R}$ is a real vector space $(A, (+, \cdot_{\mathbb{R}}))$ endowed with a bilinear product map $×: A^2 \mapsto A$.

Unary algebra.

Banach algebra.

🏷 Category=Algebra